In a bowl or zip-top bag, combine 1 Tablespoon olive oil, 1 Tablespoon balsamic vinegar, 2 cloves of minced garlic, ½ teaspoon dried oregano, ½ teaspoon kosher salt, and ¼ teaspoon black pepper.
Stir, then add chicken breasts and turn so both sides are exposed to the marinade.
Cover tightly and marinate for 15-20 minutes at room temperature, or up to 12 hours in the fridge.
When ready to bake, warm the oven to 375°F and get out a large oven-safe skillet.
Chop the tomatoes in half and combine in a bowl with the sliced shallot, pesto, 1 Tablespoon balsamic vinegar, and 2 cloves of minced garlic. Set aside.
Melt butter and warm 1 Tablespoon olive oil in the skillet. When hot, add chicken breasts and brown for just 1-2 minutes per side. It should not be cooked through.
Remove pan from the heat and scatter the tomato mixture around the pieces of chicken.
Transfer to the oven and bake until the chicken is cooked through very nearly to 165°F. Sprinkle cheese on top of each piece chicken and broil for a final 1-2 minutes. (The cooking time will vary depending on how thick your chicken is, but plan on 10-20 minutes.)
Top with fresh basil leaves, if desired, and serve.