Place the salt in a large pot of water and bring to a boil. Add the fettuccine and cook according to the package directions for al dente. During the last 3 minutes of cooking, add the broccoli. Drain in a colander, reserving 1 cup of the water; set aside. Place the butter in the pot, reduce heat to medium-low, and stir until melted. Add 1/2 cup of the reserved pasta water, then stir in 1/3 cup of the Parmesan. Add the fettuccine and broccoli and the cayenne and nutmeg, if desired; toss. Remove from heat and sprinkle with another 1/3 cup of the Parmesan and the pepper. Toss again, adding more pasta water if the fettuccine is too sticky. Serve in bowls and sprinkle with the remaining Parmesan.