Drain mozzarella on paper towels for 15 minutes and blot the cubes dry. Process with ricotta, 1 tablespoon olive oil, 1/4 teaspoon salt and pepper until a thick spread forms, stopping to scrape the bowl.
Heat remaining oil in a skillet over medium-high heat. Add whole cherry tomatoes and cook for 6 to 8 minutes, shaking the pan, until blistered and softened with a few beginning to split.
Remove the pan from heat and let tomatoes cool for 5 minutes. Lift them into a bowl, leaving loose juices behind, and fold with basil, balsamic and remaining salt.
Split the rolls and toast their cut faces under the broiler for 1 to 2 minutes. Let cool briefly, then spread the mozzarella mixture generously along both cut faces.
Divide the warm blistered tomatoes among the rolls and close gently. Cut across the middle and serve one sub per person, keeping the cheese spread inside the bread rather than pressing it flat.