To make the filling: In a bowl, combine the ricotta, mozzarella, Parmigiano-Reggiano, garlic, ¼ cup basil, ¼ teaspoon salt and ¼ teaspoon pepper.
In another small bowl, combine the tomatoes, remaining basil, ¼ teaspoon salt, ¼ teaspoon pepper and 2 tablespoons olive oil.
Put 1 chicken breast half on a cutting board and, beginning in middle of side of breast half, horizontally insert a sharp thin knife ¾ of the way through center, moving knife in a fanning motion to create a pocket. Form pocket in remaining breast halves in same manner. Divide filling between pockets and spread evenly with a finger.
Pat chicken dry and season with remaining salt and pepper.
In a large heavy ovenproof skillet heat remaining 1 tablespoon oil over moderately high heat until hot but not smoking. Brown chicken, skin-side down, about 2 minutes. Turn chicken over and brown about 2 minutes more. Transfer skillet to middle of oven and bake chicken until cooked through, about 15 minutes. Transfer chicken to a cutting board and let rest for 10 minutes.
Slice each breast crosswise into 5 slices. Divide the tomato-basil mixture among 4 dinner plates. Arrange the slices on top of the tomatoes and serve.