For the broccoli: Bring a large pot of water to a boil over medium-high heat and add the salt. Blanch the broccoli in the boiling water until bright green and crisp-tender, about 3 minutes. Drain immediately and set aside.
For the fettuccine: Fill a large stockpot three-quarters full with water and add 3 tablespoons of the salt. Bring to a boil over medium-high heat. Add the cauliflower and cook until very soft, 13 to 15 minutes. Drain and set aside.
Melt 3 tablespoons of the butter in a large skillet over low heat. Add the garlic and saute until soft and fragrant, 2 to 3 minutes. Transfer to a food processor.
Working in 2 batches, add the cauliflower to the food processor with about 1/2 cup of the milk each time. Puree until very smooth, about 1 minute.
Add the remaining 3 tablespoons butter to the skillet, then add the cauliflower mixture, nutmeg, pepper and 1 teaspoon salt. Add the heavy cream and cook over low heat until heated through, 2 to 3 minutes. Add 3/4 cup of the Parmesan and cook, stirring, until melted. Keep warm until ready to serve.
Bring a large pot of water to a boil over medium-high heat; add the remaining 1 tablespoon salt. Cook the fettuccine according to the package directions. Drain and keep warm.
For the chicken: Meanwhile, heat a grill pan over high heat.
Sprinkle the chicken on both sides with the salt and pepper. Spray the grill pan with cooking spray. Cook the chicken on both sides until golden brown and grill marks appear, 4 to 6 minutes per side. Let rest for 5 minutes, then cut into slices.
Combine the fettuccine and sauce in a large bowl, stirring until the pasta is coated. Toss with the broccoli and sliced chicken. Serve immediately, sprinkled with the remaining 1/4 cup Parmesan.