1/4 cup white-wine vinegar, plus more to taste
1/4 cup water
1 teaspoon sugar
2 cloves garlic, smashed
1 small jalapeno chile, stem, ribs, and seeds removed (optional), quartered
Coarse salt and freshly ground pepper
1 English cucumber (12 ounces), peeled, seeded, and cut into 1/4-inch dice
5 radishes, very thinly sliced
8 ounces cherry tomatoes, halved, or quartered if large
1 small red onion, cut into 1/4-inch dice
6 boneless, skinless chicken breast halves (2 pounds)
Vegetable oil, for grill
1 cup fresh mint, torn into 1/2-inch pieces, plus sprigs for garnish
Bring vinegar, water, sugar, garlic, jalapeno, and 3/4 teaspoon salt to a boil in a small saucepan. Remove from heat, and let stand for 15 minutes. Strain though a fine sieve; discard solids. Let cool completely.
Combine cucumber, radishes, tomatoes, and onion in a medium bowl. Pour in vinegar mixture, and toss to coat.
Preheat grill to medium-high. Season chicken breasts on both sides with salt and pepper. Lightly oil the grill grate. Working in batches, grill chicken until cooked through, 6 to 7 minutes per side.
Transfer to a platter, and let stand for 10 minutes. Stir mint into relish. Season with salt and pepper, and drizzle with vinegar if desired. Spoon relish on top of chicken, and garnish with mint sprigs