Preheat a charcoal grill to medium, ash-covered coals or a gas grill to 300° to 350° (medium) heat. Brush one side of bread and onion with 2 Tbsp. vinaigrette. Sprinkle onion with 1/4 tsp. salt and 1/4 tsp. pepper.
Grill steaks and onion at the same time. Grill steaks on a charcoal grill, without grill lid, 10 to 12 minutes, or on a gas grill, covered with grill lid, 7 to 10 minutes. Grill onion, on one side only, on a charcoal grill, without grill lid, 8 minutes or until tender and grill marks appear, or on a gas grill, covered with grill lid, 6 minutes or until tender and grill marks appear.
Remove steaks and onion from grill. Sprinkle steaks with remaining 1/2 tsp. salt and 1/2 tsp. pepper. Let steaks stand 5 minutes.
Meanwhile, grill bread on a charcoal grill, without grill lid, 1 to 2 minutes on each side or until lightly browned and grill marks appear, or on a gas grill, covered with grill lid, 1 to 2 minutes on each side or until lightly browned and grill marks appear.
Carve steaks into thin strips. Cut strips into bite-size pieces. Coarsely chop onion. Cut bread into 1-inch cubes.
Toss steak pieces, onion, bread, tomatoes, and basil with remaining 4 Tbsp. vinaigrette. Season with salt and pepper to taste. Sprinkle with cheese. Let stand 5 minutes before serving. Garnish, if desired.
