Bring a large pot of water to a boil and season generously with salt. Make an ice bath in a medium bowl and set aside.
To the boiling water, add the broccoli florets and cook for 2 minutes, or until the broccoli is tender and bright green.
Immediately strain the broccoli out with a slotted spoon and blanch in the ice water. Keep the water boiling to cook the pasta later.
To a food processor add the broccoli, basil, parm, lemon zest and juice, toasted pine nuts and garlic cloves. Pulse 10-12 times or until all the ingredients are finely ground and evenly combined. With the food processor running, slowly stream in the olive oil until a thick paste forms. Turn off the machine, taste and season with salt and pepper as needed. Use immediately, or store in an airtight container in the refrigerator for 2-3 days or in the freezer for up to 1 month.
When ready to serve, cook the pasta in the boiling water until al dente. Before draining, reserve 1 cup of pasta cooking liquid. Toss the pasta in the pesto, adding a small amount of pasta water if needed until you reach the desired consistency. Serve immediately with more freshly shaved parm and lemon zest.