Heat the oven to 300°F. Place tomato halves cut-side up on a lined rimmed sheet and brush with 1 tablespoon olive oil. Season with 1/2 teaspoon salt and 1/8 teaspoon pepper.
Roast for 75 to 90 minutes until softened and shrunken with lightly browned edges. Transfer to a shallow container and refrigerate promptly for about 60 minutes until cold; prep time includes chilling.
Lay mozzarella slices on paper towels for 10 minutes and blot away surface moisture. Stir chopped basil, lemon zest, lemon juice, remaining olive oil, remaining salt and remaining pepper together.
Brush the basil mixture over the cut faces of the rolls. Divide mozzarella and cold roasted tomatoes among the bottoms, leaving excess tomato liquid in the container.
Close with the roll tops and cut across the middle. Serve cold or refrigerate at 40°F or below, transporting with ice packs.