1 cup peeled, seeded and coarsely chopped tomatoes
12 small clams, scrubbed and rinsed
12 mussels—scrubbed, rinsed and debearded
3 cups shrimp stock or clam juice
12 freshly shucked oysters, liquor reserved
Put the Piri Piri Sauce in a bowl and let stand at room temperature for several hours until the oil separates. Carefully skim off 1/4 cup of the oil and transfer to a separate container. Reserve the solids.
Fill a medium stockpot three-fourths full with salted water and bring to a boil. Plunge the lobster in headfirst, cover and cook until red, about 7 minutes. Using tongs, carefully transfer the lobster to a large bowl of ice water to cool. When the lobster is cool enough to handle, twist off the tail and claws. Crack the shells and remove the meat. Discard the intestinal vein that runs the length of the tail. Coarsely chop the meat and reserve.
Bring a large saucepan of salted water to a boil. Add the pasta and cook, stirring occasionally, until al dente. Drain the pasta in a colander and return it to the pot. Add the reserved Piri Piri oil, 1/4 cup of the Parmigiano-Reggiano cheese, 2 tablespoons of the basil, 1 teaspoon of the salt and the white pepper. Toss well to evenly coat the pasta. Cover and keep warm.
Season the shrimp with the Essence and the remaining 1 teaspoon of salt. In a large skillet, heat the olive oil. Add the chorizo and cook over high heat, stirring occasionally, until browned, about 2 minutes. Add the shallots and garlic and cook, stirring, for 1 minute. Add the shrimp and cook, stirring, for 30 seconds. Add the tomatoes, clams, mussels, the reserved solids from the Piri Piri Sauce and the shrimp stock. Cover and simmer until the clams and mussels open, about 4 minutes. Add the oysters and their liquor and the chopped lobster and cook, stirring, until the oysters start to curl around the edges, about 2 minutes. Discard any clams or mussels that don't open.
Swirl the pasta in the center of 6 large plates. Use a slotted spoon to surround the pasta with the seafood. Pour the broth over the pasta, garnish with the remaining cheese and basil and serve immediately.