Bring a large pot of water to a boil. Salt it pretty liberally.
Add the broccoli florets to the boiling water and cook for 2 minutes. Scoop the broccoli out of the water and into a colander, then run cold water over the broccoli to stop the cooking. DO NOT drain the cooking water.
Return the water to a boil, if needed, and add the pasta to the water. Cook according to package directions.
Just prior to the pasta being done, scoop ½ cup of starchy pasta water out and set it aside. Drain the pasta and set aside.
Put the cooked broccoli, basil, garlic, walnuts, and cheese into the bowl of a food processor or blender. Pulse until everything is finely chopped. Scrape down the sides of the bowl.
Turn the food processor on again, and stream the olive oil in.
Slowly stream some of the starchy pasta water into the pesto until you have reached your desired consistency.
Season with kosher salt and black pepper to taste.
Toss with the cooked pasta and enjoy.