Step 1

Whisk water and 3 tablespoons salt in clean 2-quart (or larger) container until salt is dissolved. Transfer 1 cup brine to 1-quart zipper-lock bag; squeeze out air and seal bag well. Add chiles, carrots, and garlic to remaining brine in 2-quart container and press down to submerge. Cut piece of parchment paper to match surface area of vegetables; press parchment flush against surface of vegetables. Place bag of brine on top of parchment and gently press down. Cover container tightly with 3 layers of cheesecloth. Secure cheesecloth with rubber bands and place container in 50- to 70-degree F/10- to 21-degree C location away from direct sunlight.

Step 2

Let vegetables ferment for 5 to 7 days; check container daily, skimming residue from surface and pressing to keep vegetables submerged. After 5 days, taste vegetables daily until they have reached desired flavor. They should be sour and savory.

Step 3

Discard bag of brine and parchment and skim off any residue. Transfer vegetables and brine to blender and process on high speed until smooth, 2 to 3 minutes. Strain chile mixture through fine-mesh strainer, pressing on solids to extract as much puree as possible; reserve solids for other use. Rinse out blender jar. Return chile puree to blender jar; add vinegar, sugar, and 4 teaspoons salt and process on high speed until well combined, about 1 minute. With blender running, slowly sprinkle in xanthan gum to avoid clumping; process until mixture is slightly thickened, about 1 minute. Transfer sauce to mason jars or hot sauce bottles and refrigerate until ready to use. (Sauce can be refrigerated for up to 3 months).