1. Preheat grill to medium-high heat.
2. Combine first 3 ingredients in a medium bowl, stirring with a whisk until sugar dissolves. Add onion, carrot, and radishes; toss well. Let stand 30 minutes; drain.
3. Combine olive oil, garlic, thyme, 1/4 teaspoon salt, and 1/4 teaspoon pepper in a large bowl, stirring with a whisk. Stir in tomatoes; set aside.
4. Lightly coat the bread with cooking spray. Place bread on grill rack, and grill for 3 minutes on each side or until crisp. Sprinkle the steak evenly with paprika, remaining 1/4 teaspoon salt, and remaining 1/4 teaspoon black pepper. Place on grill rack coated with cooking spray, and grill for 3 minutes on each side or until desired degree of doneness. Let stand 10 minutes. Cut steak across the grain into thin slices. Cut each slice in half.
5. Cut the bread into (1-inch) cubes. Add bread, greens, steak, and basil to tomato mixture; toss well. Top with carrot mixture.