Bring a large pot of water to a boil, salt it, add the spaghetti and cook until just shy of al dente (the spaghetti will continue to cook in the sauce for a minute or two). Drain, reserving a ladleful of the pasta cooking water.
While the pasta is working, in a large, deep skillet, heat the EVOO, 4 turns of the pan, over medium heat. Add the garlic and anchovies and cook, stirring with a wooden spoon, until the garlic is tender and the anchovies are melted, about 5 minutes.
Using a food processor, process the piquillos into a paste, 1 minute. Stir the piquillos into the garlic-anchovy oil and warm through; season with pepper.
Stir the reserved pasta cooking water into the sauce. Toss the pasta and sauce together for a minute or two to allow the pasta to soak up the sauce. Toss in the olives and nuts and transfer to a platter or 4 shallow bowls.