Pat the chicken breasts dry, slice into 0.5-inch cutlets, and season with salt and pepper. Heat the 2 tbsp olive oil in a large skillet over medium-high heat and sear the chicken for 4 to 5 minutes per side until golden and cooked to 165°F (74°C). Transfer to a cutting board.

Bring a large pot of salted water to a boil. Blanch the 3 cups broccoli florets for 2 minutes, remove with a slotted spoon, and set aside. Cook the fettuccine in the same water until al dente, about 10 to 11 minutes. Reserve 0.5 cup pasta water, then drain.

In the same skillet over medium heat, melt the 2 tbsp butter. Add the 4 minced garlic cloves and cook for 1 minute. Pour in the 1 cup chicken broth and simmer for 2 minutes. Add the 1.5 cups heavy cream, 1 tsp Italian seasoning, and 0.5 tsp red pepper flakes. Simmer for 4 to 5 minutes until the sauce coats a spoon.

Reduce heat to low. Stir in the 1.5 cups Parmesan in 3 additions. Add the drained fettuccine and broccoli and toss for 1 minute. Add reserved pasta water as needed to loosen the sauce. Slice the chicken and return it to the pan with the remaining 1 tbsp butter, tossing for 1 more minute.

Divide into bowls, top with chicken slices, a spoonful of extra sauce, a fresh grating of Parmesan, and a pinch of red pepper flakes. Serve immediately.