Would you believe me if I told you that hot water, a scrub brush, and some baking soda are everything you need to keep your enameled cast iron Dutch oven clean?
Put your Dutch oven on a cooking surface and fill it with 2 cups of water. Add 4 tablespoons of baking soda.
Turn on the high heat and bring the mixture to a boil, then turn off the heat. Wait until the water reaches a temperature you are comfortable to work in.
Use a scrub brush to tackle the stains.
Get rid of the baking soda water (which is probably very dirty at this point) and fill it with warm soapy water. Clean off any remaining stains with a nylon scrubbing pad or non-abrasive sponge - just stay away from pads with metal wires or anything too abrasive.
Video
Notes
Avoid the thermal shock by never washing your enamel Dutch oven with cold water when it is still very hot. I know I was guilty of doing that often back in the day when I didn't know about the dangers of it. Essentially, when you expose your enamel coating to sudden drastic temperature changes, the material shrinks and expands quickly, which can cause cracks in the enamel finish!