Trim fat from steaks. Season steaks with salt and pepper. For the vegetable relish, cut large mushrooms into halves or quarters and thread on metal skewers. Brush mushrooms and onion lightly with olive oil.
Place steaks, mushrooms, onion, and tomatoes on the rack of the grill directly over medium heat. Grill steaks until desired doneness, turning once. (Allow 11 to 15 minutes for medium-rare doneness [145 degrees F] or 14 to 18 minutes for medium doneness [160 degrees F].) Grill mushrooms, onion, and tomatoes until tender, turning once halfway through grilling. (Allow 10 minutes for mushrooms, 15 minutes for onion, and 7 minutes for tomatoes.) Remove vegetables and steaks from grill as they are done.
Cut onion and tomato halves into 1-inch pieces. Remove mushrooms from skewers. In a medium bowl stir together basil, garlic, the 2 tablespoons olive oil, the 1 teaspoon kosher salt, and the 1 teaspoon black pepper; stir in grilled vegetables. To serve, spoon warm relish over steaks. Makes 4 servings.