Heat the 1 tablespoon of oil reserved from the anchovy can in a large deep skillet over medium-high heat.
Add the sausage (casing removed) and break it up with a spoon; cook, stirring often, until it begins to brown and crisp, about 7–10 minutes.
Stir in the chopped anchovy fillets and chopped garlic and cook for 1 minute until fragrant.
Add the crushed tomatoes, halved olives, drained capers, smoked paprika, and dried oregano; stir to combine.
Bring the sauce to a simmer and cook about 5 minutes to blend the flavors; taste and add salt and pepper only if needed, because anchovies, olives and capers can be very salty.
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 (al dente). Drain the pasta.
Divide the drained pasta among plates and top with the sausage and olive sauce; serve with grated Parmesan if desired.