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For the chicken: Rinse chicken and pat dry; place in a non metallic dish or ziplock plastic bag. Combine olive oil, basil, Romano cheese, garlic and Italian seasoning; pour over chicken, tossing to coat all sides. Cover dish or seal bag; refrigerate at least 4 hours, turning occasionally.
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Preheat charcoal or gas grill to medium-high heat. Lightly coat grill grate with oil or non-stick cooking spray. Remove chicken from marinade; discard marinade. Grill chicken about 4-6 inches from heat source for about 7 minutes per side, or until no longer pink inside (160°F on a meat thermometer).
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Lower grill to low. Top chicken with mozzarella cheese. Close gill and let cheese melt. Put chicken on a platter and top with some tomato basil butter. Serve chicken with remaining tomato basil butter on the side.
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For the tomato basil butter. Put softened butter in a small bowl. Add remaining ingredients and blend well. Season with salt and pepper; set aside.