In a large pot of boiling salted water, cook the pasta until al dente. Drain the pasta, reserving 3/4 cup of the cooking water.
Meanwhile, in a large, deep skillet, heat 1 tablespoon of the olive oil until shimmering. Add the sausage in 1-inch clumps and cook over moderately high heat, turning once, until browned and cooked through, about 8 minutes. Using a slotted spoon, transfer the sausage to a plate.
Add 2 tablespoons of the olive oil to the skillet along with the tomatoes and crushed red pepper; cook over moderately high heat, pressing the tomatoes until slightly softened, about 4 minutes. Add the reserved pasta cooking water, season with salt and cook over moderate heat, scraping up any browned bits and crushing the tomatoes, until the liquid is reduced by half and the tomatoes are nearly broken down, about 5 minutes.
Add the pasta to the skillet along with the sausage and the remaining 1 tablespoon of olive oil and cook, stirring, until the pasta is evenly coated, about 3 minutes. Remove from the heat and stir in the provolone. Transfer to bowls and serve right away.