Bring a large pot of salted water to a boil and cook the pasta until just al dente, about 8-10 minutes. Drain, reserving a cup of pasta water, and set aside.
While the pasta cooks, trim the dark green tops and root ends from the leeks, then slice them thinly. Rinse thoroughly under cold water to remove any grit between the layers.
Heat the olive oil in a large skillet over medium heat. Add the sliced leeks and cook, stirring occasionally, until they soften and begin to turn golden and fragrant, about 8 minutes.
Add the minced garlic to the skillet and cook for about 30 seconds, until fragrant and slightly golden. Stir in the lemon zest for a fresh citrus aroma.
Stir in the fresh spinach and cook for 2-3 minutes, until it wilts and turns bright green. The greens should be tender but still vibrant.
Add the cooked pasta to the skillet with the greens, tossing gently to combine. If the mixture seems dry, add a splash of the reserved pasta water to loosen and create a silky sauce.
Drizzle the dish with a little more olive oil if desired, taste, and adjust seasoning with salt or more lemon zest as needed. Toss again to evenly distribute the flavors.
Serve immediately, garnished with extra lemon zest or a drizzle of olive oil for added brightness. Enjoy this simple, earthy pasta celebrating fresh spring greens!