Slice each chicken breast horizontally to make 4 cutlets and cover with plastic wrap; pound to about 1/2-inch thickness with a meat mallet or heavy can.
In a small bowl, combine smoked paprika, garlic powder, onion powder, salt, oregano, thyme, 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 fettuccine until just al dente. Reserve 1/2 cup pasta cooking water, then drain, rinse briefly, and toss with a drizzle of olive oil; set aside.
Heat 2 tablespoons butter and 2 tablespoons olive 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 and loosely tent with foil. Wait at least 5 minutes before slicing.
Do not wipe out the skillet. Over medium-low heat, add the remaining 2 tablespoons butter to the pan drippings, then add 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 add to 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 smooth, about 2 minutes; adjust thickness with reserved pasta water as needed.
Add the cooked fettuccine to the sauce and toss to coat. Slice the rested chicken and add it to the skillet with any juices from the cutting board. Taste and adjust seasoning with salt, pepper or extra cayenne if desired, then serve.