Preheat a skillet to medium heat. Add the walnuts and gently toss until toasted, about 5 minutes.
In a food processor, pulse together walnuts and garlic cloves. Next, add in the chopped broccoli, basil, stock, red pepper flakes, 3⁄4 tsp kosher salt, 1⁄4 tsp ground pepper, and lemon juice. Pulse until mostly combined. Take the olive oil and while the food processor is running drizzle it down the feed tube, scraping down the sides of the bowl as needed. Next, add in the 2 tbsp cheese and pulse until the pesto is smooth.
Once the pesto is prepped, cook the pasta according to the package instructions. You will want to reserve 1⁄2 cup of pasta water just in case. Next, add the cooked pasta to a mixing bowl and toss with pesto. Add the pasta water as needed until the pesto has covered all the pasta.
To serve, transfer pasta to a serving bowl. Garnish with remaining cheese, walnuts, and basil. Serve and enjoy!