Bring a large pot of salted water to a boil. Prepare the pasta according to the package instructions, cooking until slightly past al dente. Drain, toss with a drizzle of olive oil to prevent sticking, and spread on a large plate or baking sheet to cool.

In a large bowl, whisk together the olive oil, lemon juice, balsamic vinegar, grated Parmesan, garlic, salt, and several grinds of pepper. Add the cooled pasta and toss to coat, then add the cherry tomatoes, mozzarella, red onion, and basil and toss again. Add the shaved Parmesan and gently toss.

Season to taste, garnish with more basil, and serve.