Toast walnuts: Bring a medium pot of salted water to a boil. Heat 1 teaspoon oil in a small skillet over medium. Add walnuts. Cook, stirring frequently, until golden-brown, about 5 minutes. Sprinkle with salt. Transfer to cutting board and let cool slightly.
Prep ingredients: Trim ends from broccoli, then cut into 2-inch florets. Peel and finely chop 1 large clove garlic. Pick basil leaves from stems, then coarsely chop leaves, discard stems. Finely grate ¼ teaspoon lemon zest, then squeeze 1 tablespoon juice, separately. Cut any remaining lemon into wedges. Finely grate Parmesan.
Cook broccoli: Add broccoli to boiling water. Cook until bright green and crisp-tender, about 3 minutes. Using a slotted spoon, transfer broccoli to a colander. Cover pot, reduce heat to low, and keep warm for step 5.
Make pesto: Finely chop walnuts and half of the cooked broccoli. Save remaining broccoli for step 6. Transfer chopped walnuts and broccoli to medium bowl. Add garlic, lemon zest, chopped basil, half of the Parmesan, and 3 tablespoons oil. Stir to combine, then season with ¼ teaspoon of salt and a few grinds of pepper.
Cook Tortelloni: Bring pot of water back to a boil. Add tortelloni and cook until al dente, about 3 minutes. Reserve ¼ cup pasta water. Add spinach to pot, then stir once, and drain tortelloni and spinach; return to pot.
Finish & serve: Add pesto, reserved broccoli florets, lemon juice, and half of the pasta water to pot. Toss gently to combine, and add more pasta water to loosen if necessary. Season to taste with salt and pepper. Divide between bowls, sprinkle with remaining Parmesan, and serve with remaining lemon wedges. Enjoy!