Lightly pound the chicken into an even thickness (about ½-inch thick; I lay a sheet of plastic wrap on top to keep things tidy). Season both sides with salt and pepper.
Dollop a tablespoon of pesto on each chicken breast, then with a brush or your fingers, slather to coat both sides. Place the cherry tomatoes in a bowl and toss with 1 tablespoon pesto. If the tomatoes seem dry, add a little olive oil.
TO GRILL: Preheat a gas or charcoal grill to high heat (~465°F). Make sure your grill is very clean and oil the grates. Place the chicken on the grill presentation (smooth) side down. Let cook for 4 to 5 minutes on the first side, then flip and cook 4 to 8 minutes on the other side, until it reaches 155-160°F on an instant read thermometer (its temperature will rise to 165°F as it rests). Grill the cherry tomatoes alongside in a grill basket or threaded onto skewers.
TO BAKE: Preheat the oven to 425°F. Line a rimmed baking sheet with parchment paper. Arrange the tomatoes around the edges of the pan and place the chicken in the center. Bake for 15 to 20 minutes, until the chicken reaches an internal temperature of 155 to 160°F (it will rise to 165°F as it rests).
Let the chicken rest for 5 minutes prior to serving with the cherry tomatoes, more pesto, and a sprinkle of fresh basil and Parmesan.
Notes
TO STORE:Refrigerate leftover pesto chicken in an airtight container for 3 to 4 days.
TO REHEAT: Reheat gently in a covered skillet over medium heat, covered in a baking dish in a 350ºF oven, or in the microwave. You can add additional pesto to keep the chicken from drying out.
TO FREEZE: Place the cooked chicken in an airtight freezer-safe storage container. Freeze for up to 3 months. Thaw in the refrigerator before reheating according to the instructions above.