Wearing gloves, trim the stem ends from the chiles and cut the flesh into large chunks (don’t remove the seeds or ribs).
Open the kitchen windows and/or turn on the exhaust fan. In a food processor, pulse the chiles, garlic, salt, and sugars to a loose, chunky paste, about 15 one-second pulses. Scrape the mixture into a glass bowl, cover loosely with plastic, and let sit at room temperature for 4 to 5 days.
Scrape the mixture into a 2-quart nonreactive saucepan. Stir in the vinegar and bring to a simmer, uncovered, over low heat, stirring frequently, until the mixture is very soft, 5 to 8 minutes.
Strain the mixture through a medium mesh sieve set over a glass bowl, pushing on the solids to extract all the liquid. (If you prefer a thicker sauce, use a large mesh sieve.) The sauce will keep, refrigerated in an airtight glass bottle or jar, for up to 4 months.
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