Bring a large pot of salted water to a boil and cook the pasta until just al dente, about 1 minute less than package instructions. Reserve a cup of pasta water before draining, then set the pasta aside.
While the pasta cooks, trim the dark green parts from the leeks and cut the white and light green sections into thin slices. Rinse the slices thoroughly under cold water to remove any grit, then drain well.
In a large skillet, melt butter over medium-low heat and add the sliced leeks. Cook gently, stirring occasionally, until they soften and turn translucent, about 15-20 minutes. The leeks should smell sweet and slightly caramelized around the edges.
Add the minced garlic to the skillet during the last 2-3 minutes of cooking, stirring until fragrant and slightly softened, about 1 minute.
Pour in the heavy cream, stirring to combine. Let it simmer gently for about 2 minutes until the sauce begins to thicken slightly and becomes velvety.
Stir in the grated Parmesan cheese and lemon zest, then season with salt and pepper to taste. Remove from heat once the cheese melts and the sauce is smooth.
Add the cooked pasta to the skillet with the sauce, tossing gently to coat evenly. If the sauce is too thick, add a splash of reserved pasta water to loosen it until glossy and smooth.
Cook the pasta and sauce together for another minute over low heat, allowing the flavors to meld and the sauce to cling beautifully to the noodles.
Taste and adjust the seasoning with more salt, pepper, or lemon zest if desired. Serve immediately, garnished with additional Parmesan if you like.
