Bring a large pot of salted water to a boil. Add the pasta and cook according to package instructions until al dente.
Meanwhile, heat the oil in a large skillet over medium-high. Add the walnuts and red-pepper flakes, if using, and cook, stirring, until golden and fragrant, about 1 minute. Using a slotted spoon, transfer walnuts and red-pepper flakes to a small bowl. Season walnuts with a little salt and pepper.
Add the broccoli to the skillet and toss to coat in the oil. Shake the skillet so broccoli settles in an even layer. Cook, undisturbed, 2 minutes. Toss and shake to arrange in an even layer again and cook, undisturbed, another 2 to 3 minutes; season with salt and pepper and remove from heat.
Drain pasta and add to the skillet along with the lemon zest, cheese, toasted walnuts and half the mint; toss to combine. Divide among plates or bowls and top with remaining mint, more cheese and a drizzle of olive oil. Serve with lemon wedges, squeezing juice on top, if desired.