Preheat a grill to medium. On a work surface, arrange the chicken cutlets in a single layer and season with salt and pepper. Divide the basil among the cutlets in an even layer, then top each with some of the mozzarella and tomatoes. Roll up the cutlets and secure with toothpicks; brush the rolls with 1 tablespoon olive oil and season with salt and pepper.
Arrange the chicken rolls on the cooking grate, cover and grill, turning occasionally, until deep golden-brown and cooked through, about 15 minutes; transfer to a work surface. Let stand for 5 minutes, then remove the toothpicks and thinly slice crosswise.
Meanwhile, in a large bowl, whisk together the remaining 2 tablespoons olive oil and the lemon juice. Add the bell pepper, cucumber and lettuce and toss; season with salt and pepper. Serve with the chicken.