Bring a large pot of salted water to a boil and cook the fettuccine according to package directions until al dente; drain and set aside.
Preheat the oven to 350°F (175°C).
Place 1/2 cup flour on a shallow plate and season the chicken breasts with 1/2 teaspoon salt and 1/4 teaspoon pepper; dredge each breast in the flour to coat evenly, shaking off excess.
In an oven-safe skillet over medium heat, melt 4 tablespoons butter. Add the floured chicken and brown on both sides about 3–4 minutes per side; it does not need to be cooked through. Remove chicken and set aside.
Discard any browned butter solids, then add 1 tablespoon olive oil to the skillet. Whisk in 1 tablespoon flour to form a paste and cook 30 seconds.
Add the minced garlic and cook 30 seconds until fragrant, then whisk in 1 cup chicken broth; the sauce will begin to thicken.
Stir in 1 tablespoon chopped thyme, 1 tablespoon capers, and 3 tablespoons lemon juice. Then stir in 1 cup heavy cream and taste; add salt and pepper if needed.
Return the chicken to the skillet, spoon some sauce over the pieces, and transfer the skillet to the preheated oven. Bake about 30 minutes, until chicken is cooked through (internal temperature 165°F/74°C).
Remove the skillet from the oven. If desired, remove the chicken to a cutting board and slice. Toss the cooked fettuccine with the sauce in the skillet until well coated.
Plate the pasta and top with the chicken, sprinkle with 1 tablespoon chopped parsley and grated Parmesan if using, and serve immediately.
