Preheat an outdoor grill for medium-high heat, and lightly oil the grate.
Place the chicken breast halves between two sheets of heavy plastic (resealable freezer bags work well) on a solid, level surface. Firmly pound the chicken with the smooth side of a meat mallet to a thickness of about 1/2 inch. Brush each chicken breast with olive oil, and sprinkle with a pinch of garlic salt.
Grill the chicken breasts on the preheated grill until the meat is no longer pink inside and the chicken has good grill marks, 5 to 8 minutes per side. Move chicken to a cooler part of the grill to keep warm.
While the chicken is cooking, mix together the tomatoes, basil, green onions, garlic, salt, black pepper, and lemon juice in a bowl.
Place an 8-inch square piece of aluminum foil onto the grill, and move a cooked chicken breast onto the foil. Top the chicken breast with about 1/4 cup of the tomato mixture, and sprinkle the topping with about 2 teaspoons of Parmesan cheese. Repeat for the other pieces of chicken.
Close the lid on the grill, and grill the chicken breasts until the topping is hot, 2 to 3 more minutes.