Slice the trimmed leeks into thin half-moons, making sure they are clean and free of grit.
Bring a large pot of salted water to a rolling boil, then add the pasta and cook until just al dente, about 8-10 minutes. Reserve about 1/2 cup of pasta water, then drain and set aside.
While the pasta cooks, melt the butter in a deep sauté pan over medium heat. Add the sliced leeks and a pinch of salt, then cook slowly, stirring occasionally, until they are soft, translucent, and fragrant, about 8 minutes.
Pour in the heavy cream, stirring gently, and warm until the mixture shimmers slightly, about 2 minutes. Be careful not to let it boil vigorously.
Add a teaspoon of fresh lemon juice and a quarter teaspoon of grated nutmeg to the sauce, stirring well to combine. Taste and adjust seasoning if needed.
Gently fold the cooked pasta into the sauce, tossing to coat evenly. If the sauce feels too thick, loosen it with a little of the reserved pasta water, a tablespoon at a time, until silky and smooth.
Remove from heat and sprinkle with chopped parsley. Give it a final gentle toss to distribute the herbs and brighten the dish with fresh flavor.
Serve immediately, spooning the creamy leek pasta into bowls and savoring the rich, tender leeks coated in silky sauce with a fresh herbal finish.