Make the cutlets: Cut the chicken horizontally to make 4 thin fillets. Then, put plastic wrap over them and gently pound to an even thickness, about half an inch, using a meat mallet or the side of a can.
Season the chicken: In a small bowl, combine all the Blackening seasoning ingredients. Set aside 1 tablespoon of this mixture for the Alfredo sauce. Use the rest of the seasoning to coat both sides of the chicken evenly.
Cook the chicken: Heat olive oil in a pan over medium-high heat, then add the chicken. Cook each side for about 4-5 minutes until it's golden brown. Take the chicken out of the pan and let it rest for 10 minutes before slicing it into strips. Leave most of the cooked bits (the "fond") in the pan, but clean off any burnt spots if there are any.
Cook the pasta in salted water according to the package instructions until it is just al dente. Reserve 1 cup of the pasta water, then drain. (This lets you adjust the sauce's thickness later on. If the sauce gets too thick, just add a bit of this water. It's also useful for reheating leftovers to keep them moist). Rinse the pasta and mix it with a little olive oil, then put it aside.
Make the sauce: Using the same skillet with the leftover drippings (don't clean it), melt the butter over medium-low heat. Add garlic and the remaining 1 tablespoon of Blackening seasoning, and cook everything together for 1 minute.
Add the flour to the skillet and whisk it in. Cook for 1-2 minutes, making sure to stir constantly.
Gradually add the half and half to the mixture in small amounts, continuously stirring as you do so.
Once the mixture starts to gently bubble, turn the heat down to low. Let it simmer while the pasta finishes cooking. As the water in the sauce evaporates, it will become thicker. Stir it from time to time.
Gradually mix the grated cheeses into the sauce, making sure to do this over low heat. After adding the cheeses, taste the sauce and season with salt and pepper as needed.
Add the drained pasta into the sauce and toss everything together until it's well mixed. The pasta will soak up the sauce, making it thicker as it cooks.
Add the chicken back into the pan and toss everything together. Let it heat up again for 1-2 minutes.
Garnish with chopped parsley and it's ready to serve!