Heat the oil in a large skillet over medium-high heat. Add the garlic and cook until fragrant and slightly golden, 1-2 minutes. Add the tomatoes and cook, stirring, until the tomatoes wilt and begin to release their water, 3-4 minutes. Season with salt and pepper and cover to keep warm.
Cook the Tortelloni according to package directions and drain, reserving 1/2 cup of the pasta cooking water. Add the Tortelloni to the tomatoes and toss with the shaved Parmigiano Reggiano cheese and basil, adding the pasta cooking water a little at a time to to loosen the sauce if desired. Season with salt and pepper, divide among bowls and garnish with more cheese and basil.