Heat the reserved 1 tablespoon of oil from the anchovy can in a large deep skillet over medium-high heat.
Add the sausage (casing removed) and break it apart with a spoon; cook until it begins to brown and crisp, about 7–10 minutes, stirring often.
Add the chopped anchovy fillets and chopped garlic to the skillet and cook, stirring, about 1 minute until fragrant.
Stir in the crushed tomatoes, halved green olives, drained capers, smoked paprika, and dried oregano.
Bring the sauce to a simmer and cook for about 5 minutes to blend the flavors; taste and add salt and pepper only if needed.
Meanwhile, cook the pasta in a large pot of boiling salted water until just tender but still firm to the bite, according to package directions.
Drain the pasta, divide among plates, and spoon the sausage and olive sauce over the top. Serve with grated Parmesan if desired.