1. Prepare the grill to medium-high heat.
2. Slice off the top ¼ inch of the tomatoes. Brush with olive oil and season with salt and pepper.
3. Place the tomatoes cut side down on the grill and cook until they soften and begin to caramelize.
4. Gently turn them over and place on a heating tray for the grill.
5. Brush the tops of the tomatoes with pesto or tapenade. Top with broken pieces of blue cheese slices. Return the tray to the grill, close top and heat until the cheese begins to soften (approx. 3 minutes).
6. Serve warm or at room temperature.
Not barbecuing? Broil the tomatoes until they begin to caramelize, then top with basil or tapenade and blue cheese. Return to the oven for about 3 minutes, just until the blue cheese softens and covers the top of the tomatoes.