Start by washing and slicing the leeks into thin rounds, making sure to remove any dirt trapped between layers. Mince the garlic and measure out the peas and broth. Zest and juice the lemon so everything is ready before cooking begins.
Bring a large pot of salted water to a boil and cook the pasta according to package instructions. Make sure it’s al dente, as it will finish cooking slightly in the sauce. Reserve about a cup of pasta water before draining.
Heat olive oil in a large skillet over medium heat and add the sliced leeks. Cook gently until they become soft and slightly sweet, but not browned. Add the garlic and cook briefly until fragrant.
Stir in the fresh peas and vegetable broth, allowing the mixture to simmer for a few minutes. This helps soften the peas and blend the flavors together. The broth also creates the base for the sauce.
Transfer the leek and pea mixture to a blender and puree until smooth. You can leave it slightly textured if you prefer a more rustic feel. Add a bit of pasta water if needed to loosen the consistency.
Return the sauce to the pan and add the drained pasta. Toss everything together gently so the pasta is evenly coated. Stir in Parmesan cheese, lemon zest, and lemon juice for added brightness.
Taste the pasta and adjust with salt and pepper as needed. The flavors should be fresh, slightly sweet, and well-balanced. Turn off the heat before adding the burrata.
Place torn pieces of burrata on top of the pasta just before serving. The creamy center will melt slightly into the warm pasta, creating a luxurious finish. Garnish with fresh herbs and serve immediately.