Bring a large pot of water to a boil, salt it, add the pasta and cook until al dente. Drain, reserving about 1/2 cup of the pasta cooking water.
While the pasta is working, in a large skillet, place the sausage in 1/2 inch of water and drizzle with EVOO. Bring to a boil, then lower the heat and simmer until cooked through and the water evaporates, 8 to 10 minutes. Transfer the sausage to a cutting board and slice into thin pieces on the diagonal. Wipe out the pan and return to high heat with another drizzle of EVOO. Add the sausage slices and cook for 2 minutes, shaking the pan occasionally.
While the sausages cook, place a large skillet over medium heat. Add 2 tablespoons EVOO, 2 turns of the pan, and the garlic and stir for 1 minute. Stir in the onion for 2 minutes. Remove from the heat. In a blender or food processor, puree the tomatoes. Add to the garlic and onion and season with salt and pepper. Stir in the basil to wilt.
Add the butter to the cooked pasta and toss. Add the tomato sauce and toss again. If the pasta sauce is too thick, add a little of the reserved pasta cooking water to loosen things up a bit. Serve the pasta in bowls and top with the cheese and sausage slices.