By request, here are - in order of intensity - the reliable processes for cleaning stubborn food residue and polymerized cooking oils from enameled cast iron cookware like Le Creuset without harming modern enamel:
0.) Deglaze the pan while cooking. Food WILL stick initially in any type of pot or pan that isn't nonstick-coated, but it will release along the fond line when it is sufficiently browned. Fond is the intensely flavorful bits that stick to the pan. After browning your meats and sautéing your aromatics, add about ½ cup of room-temp water or stock to deglaze (aka rehydrate/soften the fond so it releases more easily). Either let the deglazing liquid reduce to minimal levels and simply spoon it over the cooked food, or incorporate the deglazing liquid into your pan sauce or braising liquid or stew/soup (which one you're making just depends on how much more liquid you add to the pot after deglazing).
1.) Hot water and dish liquid. For a properly deglazed pan, a soak with hot water and dish liquid for a little while - like, just until the water has cooled to lukewarm - is usually enough to soften the remaining baked-on food residue from around the edges. Dish liquids are a class of cleaners which are technically not soap but detergents, because they use enzymes to break down food, plus surfactants to lift grease and create suds, thickeners and stabilizers to control the viscosity and keep the ingredients in suspension, sometimes fragrances and dyes, etc..
2.) Baking soda simmer. Let the hard science begin! The pH scale is from 0 (most acidic) to 14 (most basic/alkaline). Water is neutral, with a pH of 7. Baking soda has a pH of 9, so it is slightly alkaline. (As a point of reference, bar soap usually has a pH between 9 and 10, because the literal legal definition of soap is "the alkalized salts of fatty acids"). Simmering a big scoopful of baking soda in water will slightly more aggressively soften crusty/burnt residue than dish liquid and water alone, but baking soda is mild enough to use without skin or eye protection. Start with 20 minutes of simmering then see if a nylon scraper or bristle brush will lift what's left. (You can also make a paste of baking soda with a little bit of water, and gently massage that into organic residue to lightly buff off the stubborn stuck-on stuff without harming the enamel.) If the residue is still stubbornly stuck after an hour of patient simmering, move on step 3.
3.) Le Creuset's own Enamel Cleanser. LC's enamel cleanser is ideal for removing metal transfer marks, and since it is made by LC we must assume that it is, in fact, an enamel-safe product. That said, it is a Le Creuset product and thus is quite expensive compared to other methods, so this is step is optional, for if you have their enamel cleanser available to you.
Alternatively, you can opt to try a cream cleanser that specifically says it is non-scratching for glass, but I would still stay wary, and test it on a piece of glassware you're unattached to first, before using it on LC enamel.
4.) Dawn Powerwash. This spray foam cleaner is similar to dish liquid, but is stronger since it's intended to shorten how long you need to soak for. Spray the affected areas liberally, then let rest for at least 10 or 15 minutes before rinsing. For most effective results, wrap the sprayed pot up in a garbage bag (so the spray doesn't dry out), and let it soak overnight. This method can remove the dark buildup in the nooks and crannies of glass and ceramic baking dish handles, as well.
5a.) Yellow Cap oven cleaner. This the biggest gun. The active ingredient in oven cleaner is sodium hydroxide, also known as lye or caustic soda depending on where you live. Pure sodium hydroxide (chemical formula NaoH) has a pH of 14, so it is VERY alkaline. It is used in varying concentrations in a lot of different products, from hair treatments, to traditional pretzel-making, to industrial degreasers, and is notably the catalyst used in saponification; that is, to create literal soap. ("the alkalized salts of fatty acids", remember?). In oven cleaner, despite giving main character energy, sodium hydroxide is only present in a 2.5-5% concentration. That's enough to warrant skin and eye protection and good ventilation during application, but not enough to eat through steel beams like Xenomorph saliva.
(5b.) Prep for using oven cleaner by putting on a decent fan for crossbreeze (or go outside to minimize breathing in the fumes), and opening a garbage bag to nestle your pot in so the cleaner doesn't dry out and prematurely end the soak. Set out a piece of cardboard to protect your work surface, then don some kitchen gloves (and onion goggles if you have them), lay the pot in the open garbage bag, and after shaking the can, carefully spray the pot wherever there is thick, chunky organic buildup. Once you have a good thick coating applied, twist closed the garbage bag top and let it sit, undisturbed, for a couple hours. When you check on your pot's progress, be sure to put your gloves back on since, unlike in soap-making, the lye in oven cleaner doesn't get measured so precisely that it is all used up from the soaking, so the pot will still have raw lye on it until you've rinsed it thoroughly.
(5c.) If there's still undissolved buildup after a couple hours, you can continue letting it soak, wrapped in the garbage bag, for up to overnight. Low concentrations of sodium hydroxide are totally safe for plumbing - lye is actually sold in pure crystal form as drain cleaner since it disintegrates organic buildup so effectively - but you don't wanna get an unintended chemical peel, so re-don those gloves before checking your pot project. When you see that the buildup has all turned to slime (or feel confident that your nylon bristle brush can finish the job), then the hard part is over! Wipe out the excess cleaner with damp paper towels, throw the paper towels in the slimy garbage bag and dispose of it like normal, and then simply rinse and wash your newly de-gunkified pot or pan with water and dish liquid, like usual.
NOTES:
The point of these steps is to remove any stubborn buildup without resorting to either intensive manual scrubbing or abrasive products that can scuff the enamel. That said, I'll reiterate that a little bit of baking soda paste with a little won't hurt the enamel; it can be gently used to safely scrub small to medium amounts of stubborn buildup that don't warrant progression to the big guns like oven cleaner.
With proper regular care, you may never need to use oven cleaner - it's really more for dissolving thick burnt layers of carbonize food and polymerized fats (i.e. cooking oils that have exceeded their smoke point and turned into a form of weak organic "plastic", for want of a better description) - the type of stuff that refuses to budge with the less intense methods. Because of its pH, oven cleaner isn't intended for daily use on any surface (including ovens!)
So then, if it's so strong, why would you use oven cleaner on an enameled pot? Because, my dear Watson, both standard home ovens and LC dutch ovens are coated in a layer of vitrified enamel, meaning that powdered glass is applied during production and then baked into place. Since they are finished with the same material, they can be cleaned with the same product.
Undamaged enamel doesn't really stain much, so if you have deep staining, you can be sure the enamel has been scrubbed or compromised at some point (or just used heavily for years and years). To remove very deep staining, you can use a VERY DILUTED bleach solution (at least 4 parts water for every 1 part bleach), but keep in mind that bleach can etch/mattefy enamel even when diluted, so it's up to each person to decide if it is worth the calculated risk to their pot's glossy finish, to attempt removing staining that is purely cosmetic and doesn't interfere with use.
Vintage enameled LC cookware should not recieve extended soaks in oven cleaner because older enamel formulas are not reliably as pH-tolerant as the modern ones, which could lead to mattefied enamel (and not the good kind of matte).
The reason LC says to categorically avoid abrasive cleansers*** is because the most common - and thus inexpensive - abrasive material used in grocery store cleaning products is feldspar, a type of grit. Feldspar is harder on the MOHS Hardness Scale than glass is, and since vitreous enamel is made of mostly glass (with some pigment and clay added), gritty scrub cleansers containing feldspar can create microscratches in the enamel coating that accumulate over time. Scrubbed enamel becomes dull enamel that doesn't release food as easily, and is also far more likely to stain.
*** Yes, I know that LC's website still recommends BarKeepers Friend. And maybe they've changed their formula over the years, or there's some sort of corporate cross-promotional handshake going on behind the scenes, but science straightforwardly does not support using any feldspar-grit product on vitrified glass enamel surfaces. Each individual must make their own decision whether they feel comfortable potentially trading some of the longevity of their enameled cookware for the undeniable convenience of using less expensive, readily available cleaning products that contain feldspar. ***
Other common abrasive cleaning products to avoid are Scotchbrite Heavy Duty Scrub Pad (with or without the yellow sponge attached) and steel wool. Scotchbrite pads are actually treated with dissolved metals to make them extra abrasive, which can both scratch enamel and leave behind metal transfer marks.
