Trim the dark green tops of the leeks and slice them in half lengthwise, then chop into thin half-moons. Rinse thoroughly in cold water to remove any grit, then pat dry with a clean towel.
Heat the butter in a large skillet over low heat until melted and starting to foam gently. Add the sliced leeks with a pinch of salt, and cook slowly, stirring occasionally. Let them soften and turn a rich golden color, about 15-20 minutes, releasing a sweet aroma.
Push the leeks to one side of the skillet, then add minced garlic to the empty space. Sauté for about 30 seconds until fragrant and slightly golden—be careful not to burn it.
Meanwhile, bring a large pot of salted water to a boil. Add the pasta and cook until just al dente, following package instructions. Before draining, scoop out about 1/4 cup of pasta water and set aside.
Drain the pasta and add it directly to the skillet with the caramelized leeks. Pour in the heavy cream and gently stir to coat the pasta evenly. If the sauce seems thick, loosen it with a splash of the reserved pasta water, stirring until silky and smooth.
Turn off the heat, then stir in the lemon juice and a small knob of butter for extra richness. Taste and season with salt and freshly cracked black pepper as needed.
Let the pasta rest off the heat for a minute or two to allow the flavors to meld, then serve immediately. The sauce should be velvety, with tender, golden leeks and a bright citrus finish.
