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 tsp salt and 1/4 tsp pepper, then dredge each breast in the flour to coat.
In an oven-safe skillet over medium heat, melt 4 tbsp butter. Add the floured chicken and brown 3–4 minutes per side until golden but not fully cooked through; transfer chicken to a plate.
Discard any excess butter from the skillet, then add 1 tbsp olive oil. Whisk in 1 tbsp flour to form a paste and cook 30 seconds, then add the minced garlic and cook another 30 seconds.
Slowly whisk in 1 cup chicken broth until the sauce begins to thicken, then stir in 1 tbsp chopped thyme, 1 tbsp capers, and 3 tbsp lemon juice.
Stir in 1 cup heavy cream and taste; season with additional salt and pepper if needed.
Return the chicken to the skillet, transfer the skillet to the preheated oven, and bake about 30 minutes until the chicken is cooked through (internal temperature 165°F).
Remove the chicken from the skillet. Toss the cooked fettuccine with the sauce in the skillet (remove chicken first if easier), then slice or return the chicken on top.
Garnish with 1 tbsp chopped parsley and serve with grated Parmesan cheese if desired.