Fill a large pot with about 4 quarts of water and 1 tablespoon salt (we will use it to cook the asparagus and then the pasta). Bring to a boil over high heat.
Add the asparagus and cook for 2-3 minutes. Use tongs to transfer to a cutting board and roughly chop so it blends more easily. Let cool for a few minutes, then transfer to a high-speed blender.
Add pasta to the boiling water and cook according to the package instructions. Reserve about 1 cup pasta water before draining.
In the blender, add the oil, garlic, nuts, parmesan, lemon, salt, and pepper. Blend until everything is chopped up. You can add a splash of pasta water if you’re having trouble blending.
Transfer the pasta back to the pot along with the pesto and about ¼ cup pasta water to start and stir vigorously to combine, adding more pasta water as needed to create a creamy sauce.
Serve pasta topped with burrata, grated parmesan, basil, lemon zest (if using), and pepper. Drizzle your best olive oil over top and sprinkle with flaky salt (or a bit of kosher salt).