How to Make It
In a small skillet, combine 1/4 cup of the olive oil with the crushed red pepper and cook over moderate heat for 2 minutes.
Meanwhile, in a large pot of boiling salted water, cook the pasta until it is barely al dente; drain and rinse under cold water to stop the cooking. Return the pasta to the pot and stir in the warm chile oil, 1/4 cup of the cheese and 2 tablespoons of the basil.
In a very large, deep skillet, heat the remaining 1/4 cup of olive oil. Add the chorizo and cook over high heat, stirring occasionally, until the fat is rendered, about 2 minutes. Add the shallot and garlic and cook until fragrant, about 1 minute. Add the tomatoes, clams and clam juice, cover and cook until the clams just begin to open, about 5 minutes. Add the mussels, cover and cook until all the shells have opened, about 3 minutes. Stir in the shrimp, season with salt and cook until pink and curled, about 2 minutes.
Using a ladle, transfer half of the liquid to the pasta in the pot and cook over high heat, tossing, until the pasta is al dente and heated through, about 2 minutes. Divide the pasta between bowls and spoon the shellfish all around. Ladle the broth on top and garnish with the remaining 2 tablespoons of shredded basil and 1/4 cup of Parmigiano-Reggiano. Serve right away.