Fill a large pot with water, salt generously, and bring to a rolling boil over high heat.
Place oil and garlic in a small saucepan on medium heat and cook until garlic begins to turn golden brown, 1-2 minutes. Reduce heat to low and cook another 2 minutes until garlic is soft. Remove from heat and set aside.
Add kale leaves to the boiling water and cook until just tender, about 3-5 minutes. Once kale is cooked, lower the heat but do not drain the water. Use tongs to transfer the dripping kale leaves directly into a blender.
Bring the water back to a boil, add the pasta and cook according to instructions. Reserve 1 cup of the cooking water, then drain. Return pasta to the pot.
To the blender, add the oil with garlic, lemon zest, lemon juice, 1 teaspoon kosher salt, freshly ground black pepper to taste, and about a 1/4 cup of the reserved pasta water. Blend into a smooth, thick green purée. Taste, adjust seasoning with salt and pepper if needed, then blend again until very creamy.
Pour the kale purée over the pasta along with half of the grated parmesan cheese. Toss well, adding a few more splashes of pasta water, until the sauce is silky and creamy and clings to the pasta.
To serve, top pasta with fresh peas and burrata, drizzle with olive oil and remaining parmesan cheese.