Separate broccoli stalk from florets and finely chop florets. Trimy 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; remove with a slotted spoon, transfer to a bowl of ice water, then drain on paper towels
In the same pot of boiling water, add pasta and cook until al dente. Drain pasta, reserving 1 cup pasta cooking liquid
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 pesto, adding pasta cooking liquid as needed to evenly coat. Toss in reserved broccoli florets and season with salt and pepper