Bring a large pot of water to a boil and then add 3 tablespoons of salt and the broccoli to the pot. Cook until the florets turn bright green - about 5 minutes. Using a slotted spider spoon, transfer the cooked broccoli to a food processor or blender. Leave the water in the pot.
Add pasta to the boiling water and cook according to the package instructions - al dente. Reserve 1 cup of the pasta water for later.
While the pasta is cooking, add your grated parmesan, basil, garlic, oil, lemon juice, 1/4 cup walnuts, salt and pepper to the food processor. Pulse until smooth.
In a large serving bowl, toss your pasta with the broccoli pesto and 1/2 cup of the reserved pasta water. Add more if needed to make sauce smooth. (Pasta water helps your sauce bind to the pasta better).
Serve with remainder toasted walnuts, lemon zest, a drizzle of olive oil and some extra parmesan cheese if desired.
Notes
Toast your walnuts in a pan or in your oven until golden brown.