Adjust oven rack to upper-middle position and heat oven to 475 degrees. Place noodles and hot tap water to cover 2-quart microwave-safe casserole dish. Microwave until noodles are softened, 3 to 6 minutes. Discard water and dry dish. Dry noodles on clean kitchen towel.
Meanwhile, combine ricotta, 1 cup mozzarella, 1 cup Parmesan, egg, ¼ cup basil, and salt in bowl. Combine tomatoes, oil, garlic, and remaining basil in medium bowl. Spread half of tomato mixture in casserole. With short side facing you, dollop ¼ cup cheese mixture on each noodle, roll, and arrange seam-side down in casserole. Pour remaining tomato mixture over roll-ups.
Wrap dish tightly with plastic wrap, poke holes in plastic, and microwave on high until roll-ups are tender and heated through, 6 to 10 minutes. Remove plastic and sprinkle remaining mozzarella and Parmesan over roll-ups. Bake until cheese is melted and lightly browned, about 5 minutes. Let stand 5 minutes. Serve.
