Bring a large pot of water to a boil and salt it lightly. Cook the pasta until al dente, then reserve 1/2 cup (120ml) of the cooking water and drain.

Heat 2 tablespoons olive oil in a large skillet over medium-high heat until it shimmers. Add the dry capers and fry for 2 to 3 minutes, until they burst open, the sizzle quiets, and they look crisp and slightly golden. Lift them out with a slotted spoon and set aside.

Lower the heat to medium. Add the garlic, red pepper flakes, and anchovy if using. Stir for 30 to 60 seconds, until fragrant and the anchovy melts into the oil. Do not let the garlic brown.

Pour in the cream and bring it to a gentle simmer, with small bubbles at the edges. Cook for 3 to 4 minutes, until it thickens slightly and lightly coats the back of a spoon.

Stir in the olives and reduce the heat to low. Remove the pan from the heat and stir in the Parmesan and lemon zest until melted and smooth.

Add the pasta and toss, adding reserved water a tablespoon at a time until the sauce clings to the strands in a glossy coat. Taste and season with black pepper, adding salt only if needed.

In a separate nonstick skillet, heat 1 tablespoon olive oil over medium-high heat. Crack in the eggs and fry for 2 to 3 minutes, until the whites are set with crisp, lacy edges and the yolks still jiggle.

Divide the pasta among bowls, top each with an egg and some of the crisp capers, and finish with parsley and extra Parmesan.