Cut the broccoli into 1-inch florets and set aside. Cut the chicken breast into 1/2-inch cubes and season generously with salt and pepper—this allows the seasoning to penetrate the meat as it sits. Mince the garlic fresh and set it aside. Fill a large pot with salted water (it should taste like the sea) and bring it to a rolling boil over high heat. I like to use fresh, finely minced garlic rather than pre-minced because the flavor is noticeably brighter and more vibrant in this delicate sauce.
Add the broccoli florets to the 3 tablespoons of hot olive oil in a large skillet over medium-high heat. Cook for 4 minutes, stirring occasionally, until the florets begin to soften and develop light browning. Add 1/4 cup water to the skillet, cover, and cook until the water evaporates (about 2 minutes). Transfer the cooked broccoli to a clean bowl. Without cleaning the skillet, add the seasoned chicken cubes to the same pan and cook over medium-high heat for 5-7 minutes, stirring occasionally, until the chicken is cooked through and lightly golden. Add the cooked chicken to the bowl with the broccoli.
Once the salted water from Step 1 is at a rolling boil, add the fettuccine and cook according to package directions until al dente (usually 9-11 minutes). Before draining, scoop out and reserve 1 cup of the starchy pasta water—this will help adjust the sauce consistency later. Drain the pasta in a colander but do not rinse it, as the surface starch helps the sauce cling to the noodles.
Return the same skillet to medium heat and add the 4 tablespoons of butter. Once melted and foaming, immediately add the minced garlic and cook for just 30 seconds, stirring constantly—this prevents the garlic from burning and keeps it sweet. Pour in the 2 cups of heavy cream and bring to a gentle simmer over medium heat, stirring occasionally. Add the 1.25 cups of grated parmesan cheese and the 1/4 teaspoon of ground nutmeg, stirring constantly until the cheese melts completely and the sauce becomes smooth and creamy (about 2-3 minutes). I prefer Kerrygold butter because its higher fat content creates a richer, silkier sauce that coats the pasta beautifully.
Add the cooked fettuccine from Step 3 to the Alfredo sauce in the skillet, followed by the chicken and broccoli mixture from Step 2. Toss everything together gently but thoroughly over medium heat for 1-2 minutes until everything is well coated and warmed through. If the sauce seems too thick, add reserved pasta water a splash at a time while tossing, until you reach your desired consistency—the pasta should move freely but be generously coated. Taste and adjust seasoning with salt and pepper if needed. Serve immediately in warm bowls, and pass additional grated parmesan cheese at the table if desired.