Cut a pocket horizontally in each chicken breast. Season lightly inside and out with salt and pepper. Place 1 cheese slice, 2 tomato slices and 2 basil leaves inside each pocket. Wrap 1 prosciutto slice crosswise around each breast to enclose the filling.
Preheat a grill to medium. Brush the wrapped chicken with 1 tbsp. EVOO and place on the grill. Cook, turning once, until well marked on both sides, about 10 minutes. Move to a cooler part of the grill and cook until no longer pink in the center, about 5 minutes. Add the lemon halves to the grill and char briefly.
Meanwhile, in a small bowl, combine the pesto and the remaining 3 tbsp. EVOO. Transfer the chicken to a platter, top with the pesto mixture and serve with the lemons.
Bonus: We challenged comedian Wendi McLendon-Covey to make this recipe! See how she did.