Make the roasted tomato tapenade: Preheat the oven to 375 degrees F and line a baking sheet with parchment paper. Halve 4 tomatoes and sprinkle with 1 teaspoon salt and a few grinds of pepper. Place the tomatoes cut-side down on the baking sheet, tucking 1 thyme sprig under each. Drizzle with 2 tablespoons olive oil and roast until the skins start to loosen, about 15 minutes. Remove from the oven; set aside until cool enough to handle, then pinch off the skins and discard. Return the tomatoes to the oven and continue roasting until slightly dried, about 1 hour. Discard the thyme and set the tomatoes aside to cool. (Keep the oven on.)
Puree the roasted tomatoes, olives, anchovy, garlic, capers, vinegar, 3 tablespoons olive oil and 2 tablespoons water in a food processor until smooth. Season with salt and set aside. (The tapenade can be made 1 day ahead; transfer to a bowl, cover and refrigerate.)
Heat a large nonstick skillet over medium-high heat and add the remaining 1 tablespoon olive oil. Working in batches, add the prosciutto slices in a single layer and cook, turning once, until golden brown and crisp, about 5 minutes. Drain on paper towels.
Cut the baguette in half crosswise, then split each half open to make 4 pieces. Place cut-side up on a baking sheet and bake until golden around the edges, about 5 minutes. Slice the remaining tomatoes. Spread the roasted tomato tapenade on the baguette. Arrange the mozzarella and sliced tomatoes on top. Season with salt and pepper, then top with the prosciutto and some arugula and basil.
Cutting the lobsters in half before cooking adds extra flavor to the broth. If you keep them whole, cook them for 2 to 3 minutes longer before adding the mussels.