Bring a large pot of salted water to a boil and cook the pasta until just al dente, about 8 minutes. Drain, reserving a cup of pasta water, and set aside.
While the pasta cooks, slice the leeks in half lengthwise, then chop into thin half-moons. Rinse thoroughly under cold water to remove any grit, then drain and set aside.
Heat the olive oil in a large skillet over medium heat until shimmering. Add the sliced leeks and a pinch of salt, then cook gently, stirring occasionally, for 10-12 minutes until they become soft, fragrant, and slightly golden.
Add the anchovy fillets to the skillet and stir until they melt into the leeks, about 2-3 minutes. The mixture will turn savory and fragrant, with a glossy sheen.
Pour in the heavy cream, stirring to combine. Bring to a gentle simmer and cook for 2 minutes until the sauce begins to thicken slightly.
Add the cooked pasta directly into the skillet, tossing gently to coat the noodles evenly with the sauce. If the sauce seems too thick, loosen it with a splash of the reserved pasta water, stirring until silky.
Stir in the grated Parmesan cheese and a squeeze of lemon juice to brighten the flavors. Toss again to distribute evenly, and cook for another minute to meld all the flavors together.
Once everything is well combined and heated through, transfer the pasta to serving plates. Garnish with extra Parmesan or herbs if desired, then serve immediately while hot and creamy.
