Start by boiling salted water and cooking your pasta until al dente. Be sure to reserve about a cup of pasta water before draining. This starchy water is key to creating a smooth sauce later. Rinse lightly if needed and set aside.
Melt butter in a large skillet over medium-low heat. Add sliced leeks, garlic, and lemon zest, cooking until soft and slightly sweet. This step builds the base flavor, so don’t rush it—about 6 minutes works best.
Stir in the peas and cook briefly for 2–3 minutes. You want them tender but still bright green. Overcooking can dull their color and sweetness, so keep an eye on them.
Transfer half of the vegetable mixture to a blender. Add parsley, olive oil, and reserved pasta water, then blend until smooth. This creates a creamy, vibrant green sauce without needing heavy cream.
Return cooked pasta to the skillet with the remaining vegetables. Pour in the blended sauce and toss everything together. Add extra pasta water if needed to loosen the sauce and coat the pasta evenly.
Sprinkle grated Parmesan over the pasta and gently mix. Tear the burrata into pieces and place it on top, letting it melt slightly into the warm dish. This final step adds a luxurious, creamy finish.