Bring a large pot of salted water to a boil. Add the broccoli florets and cook them for 1 minute. Using a slotted spoon, transfer the broccoli to a blender.
Add the pasta to the boiling water and cook until al dente, 10 to 12 minutes. Drain the pasta, reserving 1 cup of the pasta water, and set it aside.
Add the nuts to a small saucepan set over medium heat. Toast the nuts, moving them around in the pan, until they are fragrant, about 3 minutes.
To the blender, add the nuts, basil, garlic, Parmesan cheese, pepper and 1/2 cup of the reserved pasta water. Pulse until combined.
With the blender running, stream in the olive oil. Stop the blender and scrape down the sides as needed then continue blending the pesto until it is puréed. (If needed, added 1 additional tablespoon of pasta water at a time to thin the pesto.) Taste and season it with salt and pepper.
Toss together the pasta and pesto, thinning it with additional reserved pasta water 1/4 cup at a time as needed. Serve.
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