Bring a large pot of salted water to a boil. Add the fettuccine and cook until al dente, according to package instructions.
About 3 minutes before the pasta is done, add the broccoli florets to the same pot. Let them cook together until the pasta is ready. Reserve ½ cup of the pasta water, then drain both the pasta and broccoli and set aside.
While the pasta is cooking, place the chicken in a medium bowl. Season with kosher salt and black pepper, tossing to coat evenly.
Heat the olive oil in a large skillet over medium-high heat. Add the chicken in a single layer and cook for 3-4 minutes without stirring, until golden brown. Flip and cook for another 2-3 minutes until cooked through. Do not overcrowd the pan. Work in batches if necessary. Transfer the chicken to a plate and set aside.
Reduce the heat to medium. In the same skillet, melt the butter. Add the minced garlic and cook for 30 seconds, stirring frequently.
Pour in the heavy cream and bring to a gentle simmer. Slowly stir in 1 cup of the Parmesan cheese until the sauce is smooth and creamy.
Return the chicken, pasta, and broccoli to the skillet. Toss gently to coat everything in the Alfredo sauce. If the sauce feels too thick, add a splash of the reserved pasta water until it reaches the desired consistency.
Serve warm, sprinkled with the remaining ¼ cup Parmesan cheese.