Pat the chicken dry with a paper towel and pound each breast, so it is 1 inch thick. The breasts should be an even thickness so they cook evenly.
Combine the oil, salt, garlic powder, dried herbs, and black pepper in a bowl.
Add the chicken to the bowl with the oil and seasonings. Toss to coat, then leave the chicken to sit while you make the basil.
Add all the pesto ingredients except the olive oil to a blender or food processor.
Blend/process until the ingredients come together.
Run the blender or food processor on low as you deliver the oil in a steady stream.
Preheat your Z-Grills (preferably with branded apple wood pellets) or other wood pellet grill to 350 degrees Fahrenheit.
Transfer the chicken to the grates, close the lid, and cook for 4 minutes.
Flip and cook for another 4 minutes or until the internal temperature reaches 160 degrees Fahrenheit.
Top each chicken breast with a slice of cheese, close the lid, and grill for another minute or two. Feel free to place the chicken on a grill mat to prevent mess.
Plate each chicken breast, then divide the pesto between each breast. Follow with tomatoes, capers (if desired), and garnish with fresh basil.