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