Make the chicken: Using a knife, butterfly the chicken and pound thin between 2 layers of plastic wrap. Transfer the chicken to a shallow dish, drizzle with olive oil, and sprinkle with the rosemary. Season the chicken with salt and pepper and refrigerate for at least 2 hours and up to 2 days. When ready to cook, preheat a grill. Grill the chicken, turning once, until just cooked through.

Make the tomatoes: Preheat the oven to 250 degrees F.

Using a small knife, score the ends of the tomatoes, drop them in a pot of boiling water, and cook briefly, about 20 seconds. Using a slotted spoon, transfer the tomatoes to a bowl of ice water to stop the cooking. Peel off the tomato skins and cut the tomatoes in 1/2 lengthwise. In a small bowl, toss the tomatoes with olive oil and season with salt and pepper. Arrange the tomatoes on a baking sheet, cut-side up, and bake until lightly browned and somewhat dried, about 3 hours.

Make the sandwiches: Preheat an electric sandwich press or grill. Layer the chicken, tomato, and mozzarella between the bread, sprinkle with rosemary and season with salt and pepper. Put the sandwiches in the press and cook until hot and crispy.