Bring a big pot of salted water to a boil. Cook the pasta following the package directions, just a touch past al dente.
Drain, drizzle a little olive oil on the pasta to keep it from sticking, and spread it on a sheet pan or large plate so it cools down.
In a large bowl, whisk the olive oil, lemon juice, balsamic vinegar, grated pecorino, garlic, salt, and a few turns of black pepper.
Add the cooled pasta to the bowl and toss it gently so every piece gets coated.
Stir in the cherry tomatoes, mozzarella, red onion, and basil. Toss again.
Add the shaved pecorino and give it a light toss one last time.
Taste, adjust seasoning if needed, top with more basil, and serve.
