Preheat oven to 375°.

In a large skillet over medium heat, heat oil. Add cherry tomatoes and balsamic and cook until tomatoes begin to break down, about 5 minutes. Season with salt, then add half of the garlic and stir until fragrant. Transfer to a bowl and wipe out skillet.

Make sauce: To same skillet over medium heat, add butter. Let melt, then add remaining garlic and cook until fragrant, 1 minute. Add flour and cook until golden, 2 minutes more. Whisk in milk and simmer until thick and creamy, 3 minutes, then stir in 1 cup mozzarella and all of the Parmesan. Spoon a thin layer of sauce onto the bottom of an 8"-x-8" baking dish.

In a small bowl, stir together ricotta and egg, then season with salt and pepper.

Assemble lasagna rolls: Lay cooked noodles in a single layer on a baking sheet. Spread with ricotta and tomato mixture, then drizzle with sauce, sprinkle with basil, and roll up. Pack the roll-ups tightly and upright in the baking dish. Spoon the rest of the sauce on top of the roll-ups and sprinkle with remaining mozzarella

Bake until golden, 27 to 30 minutes.