In a small bowl, combine the cherry tomatoes, sun-dried tomatoes, capers, garlic, sherry vinegar, 1 tsp. of the thyme, 3 Tbs. of the olive oil, 1/4 tsp. salt and a few grinds of pepper. Set aside, stirring occasionally to let the flavors combine.
Heat a gas grill to high. Put the chicken in a shallow nonreactive pan or plate. Rub the chicken all over with salt (about 1/2 tsp. for each piece of chicken), a few grinds of pepper, and the remaining 1 tsp. thyme. Drizzle 1/2 tsp. olive oil over each side of each piece of chicken and rub all over.
Lay the butterflied breasts on the hot grill grates and cook, covered, until the chicken has grill marks, 1-1/2 min. With tongs, rotate the breasts 90 degrees (to get a crosshatch of grill marks) and continue grilling until grill marks form, another 1-1/2 min. Flip the breasts and cook the second side in the same way but for a little less time, grilling for 1 min. in one direction and 1 min. in another, until cooked through.
Transfer the breasts immediately to two warm serving plates, stir the tomato mixture and spoon equal amounts (it will be a generous portion) over each piece of chicken. Garnish with a sprig of thyme, if you like.