How to Make It
Cut off and reserve the broccoli florets. Peel the broccoli stalks and cut into 1/2-inch circles.
In a food processor with the motor running, drop in the garlic and finely chop. Add the nuts, cheese, 1 teaspoon salt and 1/2 teaspoon pepper and pulse to finely chop. Add the broccoli florets and pulse to finely chop. With the motor running, add the oil, blending to combine.
In a large pot of boiling salted water, cook the pasta until al dente. 5 minutes before the pasta is done, add the reserved broccoli stalks. Reserve 3/4 cup of the pasta cooking water, then drain the pasta and broccoli stalks. Toss the pasta with the pesto and about half the reserved cooking water. Add more water for a looser consistency. Season with salt and pepper to taste, then serve.