Bring a large pot of salted water to a boil.
In a food processor, combine the basil leaves, toasted pine nuts, garlic, and Parmesan cheese; pulse until finely chopped.
With the processor running, slowly stream in the extra-virgin olive oil until the mixture is emulsified and smooth.
Season the pesto with lemon juice, salt, and black pepper; set aside.
Add the chickpea pasta to the boiling water and cook for 5 minutes.
Add the broccoli florets to the pasta pot and continue boiling for an additional 3 to 4 minutes until the pasta is al dente and broccoli is tender-crisp.
Reserve 1/2 cup of the pasta cooking water, then drain the pasta and broccoli.
Return the pasta and broccoli to the pot over low heat; add the cannellini beans and the prepared pesto.
Toss thoroughly to coat, adding the reserved pasta water 1 tablespoon at a time until the sauce reaches a creamy consistency.
Adjust seasoning with salt and pepper to taste and serve immediately.