In a large sauté pan, heat the olive oil over medium-low heat. Add the garlic and cook until fragrant, about 1 minute.
Add the cherry tomatoes, Roma tomatoes, and heirloom tomatoes. Season with salt and black pepper. Stir to combine.
Reduce the heat to low and simmer gently for 30 minutes, stirring occasionally, until the tomatoes have completely softened and broken down.
Bring a large pot of salted water to a boil and cook the paccheri until just shy of al dente. Reserve 1 cup of pasta water before draining.
Remove and discard the garlic. Transfer the tomato mixture to a blender and blend until completely smooth.
Pass the sauce through a fine-mesh sieve and return it to the sauté pan.
Simmer for 2-3 minutes to slightly reduce and concentrate the flavor.
Add the drained paccheri to the sauce along with a ladle of reserved pasta water, the butter, and the fresh basil.
Toss gently over low heat until the butter emulsifies into the sauce and the pasta is coated in a silky, glossy finish.
Remove from the heat and fold in grated Parmigiano Reggiano.
