1. Process carrot in a food processor until very finely chopped; set aside. Process onion in food processor until very finely chopped; set aside. Pulse chiles in food processor. Return carrot and onion to food processor. Add mangoes, salt, and sugar. Pulse to puree, about 2 minutes.

2. Transfer to a medium saucepan, and stir in vinegar and 3/4 cup water. Bring to a boil. Reduce to a simmer; cook, stirring occasionally, about 25 minutes. Let cool completely. Divide among small bottles, or pour into an airtight container. Store in the refrigerator up to 6 weeks.