Boil the water. Bring a large stockpot of generously-salted water to a boil over high heat.
Cook the broccoli. Once the water is boiling, add the broccoli and stir to combine. Cook for 3 minutes or until tender. Use a spider strainer or a slotted spoon to transfer the broccoli to a food processor, leaving the remaining water boiling in the stockpot.
Cook the pasta. Add the pasta to the stockpot and cook until *just* al dente.
Prepare the pesto. Add the basil, Parmesan, pine nuts, spinach, garlic, lemon zest, lemon juice, salt and pepper to the food processor. Pulse to blend while slowly drizzling in the olive oil until the mixture is smooth. Scoop out ΒΌ cup of the starchy boiling pasta water and add it to the pesto. Pulse until super-smooth and creamy. (If needed, add extra pasta water to achieve that super-smooth consistency.)
Toss. Once the pasta is al dente, scoop out an additional 1/2 cup of the starchy boiling pasta water and set it aside. Strain the pasta and return it to the stockpot. Add the broccoli pesto and toss to combine. If the pesto seems at all dry, add in a bit of the starchy water and toss to combine.
Serve. Serve warm, garnished with extra Parmesan and a twist of black pepper, and enjoy!