1. Presoak bamboo skewers for 30 minutes or have ready metal skewers. Slide the tomatoes on skewers. Set aside.
2. Heat a gas or electric grill to medium heat and generously brush the cooking grate with oil.
3. Place the broccoli rabe in a colander and wash well under cold running water. Run a knife down the center of any thick stalks, more than ½-inch thick, leaving the stems intact at top. Drizzle with olive oil and season with salt and pepper to taste. Set aside.
4. Brush the lemons with ½ tablespoon olive oil and season with salt and pepper to taste. Place the sugar on a small plate and dip each of the lemon halves, cut-side down, into the sugar. Grill the lemons cut-side down until golden brown at the edges, about 5 minutes. Using tongs, carefully flip the lemons over and grill for 2 more minutes. Remove from grill and set aside. When cool enough to handle, squeeze the lemon juice into a small bowl.
5. Generously brush both sides of the bread slices with olive oil and season with salt and pepper to taste. Grill the bread, flipping once until crisp and golden brown, about 4 minutes per side. Transfer to a plate and allow to cool.
6. Meanwhile, arrange the broccoli rabe in a single layer across the grill grate. Grill, turning often, for about 4 minutes. Transfer to a large bowl.
7. Brush the tomatoes with ½ tablespoon of olive oil and season with salt and pepper to taste. Arrange the tomato skewers evenly across the grill grate. Grill until nicely blistered in spots, about 5 minutes. When cool enough to handle, carefully remove the tomatoes from the skewers into the bowl with the broccoli rabe.
8. Cut bread slices into ½-inch cubes and add to the bowl with the vegetables. Add the basil, garlic and 3 tablespoons of lemon juice, tossing until well combined. Arrange on a decorative platter. Drizzle with additional olive oil and lemon juice, if desired. Top with cheese. Adjust seasonings with salt and pepper to taste. Serve at room temperature.