Slice each chicken breast horizontally to make four cutlets, cover with plastic wrap, and pound to about 1/2-inch thickness using a meat mallet or the side of a can.
In a small bowl, combine the smoked paprika, garlic powder, onion powder, salt, oregano, thyme, black pepper, and cayenne; reserve 1 tablespoon of this blackening seasoning for the sauce and use the rest to season both sides of the chicken.
Bring a large pot of heavily salted water to a boil and cook the fettuccine until just al dente. Reserve 1/2 cup pasta cooking water, drain, rinse briefly, and toss with a drizzle of olive oil; set aside.
Heat 2 tablespoons butter and 2 tablespoons oil in a large heavy-bottomed skillet over medium-high heat. When hot, add the seasoned chicken and cook 4–5 minutes per side until golden and cooked through. Transfer to a cutting board, loosely tent with foil, and rest at least 5 minutes.
Do not wipe the skillet. Reduce heat to medium-low and melt the remaining 2 tablespoons butter in the pan drippings. Add the minced garlic and the reserved 1 tablespoon blackening seasoning and cook about 1–3 minutes until fragrant.
Add about half of the chicken broth to the skillet, then whisk the remaining broth with 1 tablespoon cornstarch until smooth and pour into the pan along with the heavy cream. Bring to a simmer and cook until slightly thickened, stirring occasionally.
Reduce heat to low and stir in the grated Parmesan until melted and the sauce is smooth, about 2 minutes. If the sauce is too thick, thin with reserved pasta water a splash at a time to reach desired consistency.
Add the cooked fettuccine to the skillet and toss to coat in the Alfredo sauce. Slice or chop the rested chicken and add it with any accumulated juices to the pan; toss gently to combine. Taste and adjust seasoning with salt, pepper, or cayenne as desired.
Serve immediately, garnishing with chopped fresh parsley if desired.