Bring a large pot of salted water to a boil. Add pasta and cook according to package directions. During the last 2 to 3 minutes of cooking, add broccoli. Reserve 1 cup pasta water; drain.
Meanwhile, heat a large skillet over medium heat. Add walnuts and cook, stirring often, until lightly toasted and fragrant, 3 to 5 minutes. Transfer walnuts to a plate.
In the same skillet, heat olive oil over medium heat. Add garlic and cook until lightly golden, about 1 minute. Stir in sweety drop peppers and red pepper flakes; cook for 30 seconds.
Add pasta and broccoli to the skillet. Toss to coat, adding reserved pasta water a little at a time until a light sauce forms.
Remove from heat. Add lemon juice, walnuts, and Parmesan; toss until combined and glossy. Taste; season with salt, pepper, red pepper flakes, and additional lemon juice to taste.
