Heat an outdoor grill or indoor grill pan to high. Season chicken with salt and pepper and drizzle with EVOO to keep it from sticking to the grill. Cook 3-4 minutes on each side and transfer to a foil-covered platter to hold. If you are using a grill pan, cook the chicken in batches if necessary.
While chicken cooks, make sauce: place a medium pot on the stove over medium heat. Add two tablespoons EVOO, about two turns of the pan. Add garlic, crushed red pepper flakes and chopped onions. Cook 10 minutes, stirring often. Add tomatoes and heat through, two minutes. Wilt in the basil and season the sauce with salt and pepper.
Pre-heat the broiler to high.
Layer the chicken with fire roasted sauce in a casserole dish. Top the casserole with Parmigiano Reggiano and mozzarella. Brown the casserole under the broiler three minutes.