This is a great recipe to sneak an extra serving of vegetables onto the plates of picky eaters. It also can be eaten or cold, making it an excellent school lunch option for kids.
Separate broccoli stalk from florets and finely chop florets. Trim woody end of stalk, then peel and finely chop. Cook stalk in a large pot of boiling salted water until bright green, about 3 minutes. Add florets and cook until crisp-tender, about 3 minutes more. Using a slotted spoon or mesh spider, transfer broccoli to a large bowl of ice water, then drain on paper towels.
Add pasta to same pot of boiling water and cook until al dente. Drain pasta, reserving 1 cup pasta cooking liquid, and return to pot.
While pasta cooks, transfer broccoli (reserve a handful of florets) to a food processor and add Parmesan, basil, oil, and lemon juice. Season with salt and pepper, then process until smooth.
Toss pasta with broccoli pesto, adding more pasta cooking liquid as needed to coat evenly. Toss in reserved broccoli florets and season with salt and pepper.
Do Ahead: Broccoli pesto can be made up to 1 day ahead. Let cool, then press plastic wrap directly on surface before covering to avoid discoloration and chill.