In a large bowl, whisk 1/4 cup of the olive oil with the lemon zest, lemon juice, thyme, garlic and a generous pinch each of salt and pepper. Spoon 1 1/2 tablespoons of the mixture into a medium bowl, add the tomatoes and turn to coat. Add the chicken to the large bowl and turn to coat. Let the tomatoes and chicken stand at room temperature for 30 minutes.

Light a grill. Brush the rolls with the remaining 2 tablespoons of olive oil and grill over high heat until golden. Add the chicken to the grill, season with salt and pepper and grill over high heat, turning once, until cooked through and lightly charred, about 6 minutes. Place the chicken on the rolls and top with the mozzarella, tomato and basil. Spoon any tomato juices on the top half of the bun, close the sandwiches and serve.