Bring a large pot of salted water to boil over high heat. Once boiling, add pasta and cook for one minute LESS than the package instructions. You will see a very tiny white core when cut in half.
While the pasta cooks, trim the bottom of the tomatoes, core them and slice in half through the core. Using the large holes on a box grater, grate the trimmed side of the tomato into a large bowl until nothing but the skin remains. Discard the skins.
Transfer the pasta to a colander to drain for about 5 minutes. While the tomatoes drain, heat a sauce pan to medium high heat and add the olive oil and garlic and cook stirring frequently until the garlic is fragrant-about 30 seconds.
Add the grated tomatoes and season with a large pinch of salt. Add 2 of the basil sprigs-stem and leaves and lower the heat to simmer for 3 to 5 minutes. Taste for seasoning adjustments and remove the sprigs of basil.
Turn off the heat and add the pasta and half the Parmesan. Using tongs, toss the pasta until is coated in the sauce and the Parmesan is melted.
Chop the remaining basil leaves and add them to the saucepan along with the remainder of the Parmesan, toss to combine.
Divide among 4 bowls, spooning over any tomato liquid left in the saucepan. Top with more Parmesan if desired and serve immediately.