Bring a large pot of water to a boil, add a pinch of salt, then stir in the fettuccine. In the last 3 minutes of cooking, add broccoli florets so they steam alongside pasta.
While pasta cooks, cut chicken into bite-sized pieces and sprinkle with salt, pepper, and garlic powder to evenly coat.
Heat a pan over medium heat with butter and oil, add the chicken in a single layer, and cook for about 5 minutes until edges turn golden and juices run clear.
Reduce heat to low, add butter, heavy cream, and Parmesan to the same pan, stirring gently until the cheese melts into a smooth sauce.
Taste the sauce and add salt if needed, then grate a small pinch of nutmeg in and stir to release its warm aroma.
Drain pasta and broccoli, reserving half a cup of pasta water. Add them into the sauce, return the chicken, and toss gently to coat. If it feels dry, splash in some reserved water.
Taste and add more salt as needed. Serve the chicken broccoli fettuccine hot with extra grated Parmesan on top.