Begin boiling the noodles in a pot. If you have a steamer to place on top of the pot, place it, pour in the frozen broccoli, and begin steaming the broccoli. If not, steam the broccoli in another pan by adding a half inch of water to the bottom of the pan and bringing to a boil. Pour in the broccoli and cover for 5-10 minutes until fork-tender. Alternatively, if you have fresh broccoli, place it on a baking sheet, drizzle with olive oil, salt, and pepper and bake at 400F for 20 minutes.
Once the noodles are done, reserve 1/2 cup of the pasta water and drain the noodles. Add the noodles to the butternut sauce along with the pasta water and stir until creamy and coated. Add in the broccoli.