Heat coals or gas grill for direct heat. Spray bell pepper pieces and onion slices with cooking spray; sprinkle with garlic powder. Place vegetables on grill. Cover and grill over medium heat 5 to 8 minutes, turning once, until tender; remove from grill. Cut pepper pieces in half.
Brush one side of each slice of bread with 1 teaspoon oil. Spread other side of 4 slices of bread with 1 tablespoon cheese. Top each with 2 basil leaves, 1 onion slice, 2 pepper pieces and turkey. Top with another slice of bread, oil side up.
Grill sandwiches over medium heat 2 to 4 minutes, turning carefully once, until bread is toasted.