How to Make It
Light the grill or heat the broiler. In a blender or food processor, puree the jalapeños, smashed garlic cloves, the 1/2 cup cilantro leaves and stems, the fish sauce, cooking oil, sesame oil, and 1/4 teaspoon of the salt. Put the chicken in a shallow dish and coat it with the cilantro puree.
Grill the chicken over moderately high heat or broil it for 5 minutes. Turn and cook until just done, about 5 minutes longer.
Meanwhile, in a small stainless-steel saucepan, bring the vinegar, sugar, and the remaining 1/4 teaspoon salt to a simmer, stirring. Simmer for 2 minutes. Pour the liquid into a small glass or stainless-steel bowl and let cool. Add the minced garlic, the 1 tablespoon chopped cilantro, the red-pepper flakes, and water. Serve each chicken breast with a small bowl of the dipping sauce alongside.
Asian fish sauce is available at Asian markets and many supermarkets.