Place peppers, garlic, sugars, and salt in a food processor, and pulse until roughly chopped. Transfer mixture to a clean container, cover, and let sit at room temperature. A mason jar with a loose-fitting lid or a clean bowl with plastic wrap will work. Fermentation should begin in about 2 days. When bubbles begin to form, stir your "mash" once or twice a day to combine and help settle. Continue until the mixture stops bubbling, about 6 to 8 days.
Transfer the mash to a blender, add the vinegar, and puree until very smooth. Strain the sauce through a fine sieve. Add xanthan gum, and puree again until it's smooth and thick.