Prepare the chicken: Place the chicken breasts between two sheets of plastic wrap and pound the thicker ends until the breasts are an even thickness. Pat dry with paper towels.
Season the chicken: Sprinkle the chicken with Italian seasoning, paprika, ½ teaspoon of salt, and ½ teaspoon of black pepper.
Cook the chicken: Heat the olive oil and 2 tablespoons butter in a large skillet set over medium-high heat. Add the chicken and cook until golden brown and cooked through, about 5 to 6 minutes per side, or until the internal temperature reaches 160 to 165°F as measured by a digital thermometer. Transfer to a clean cutting board, loosely cover, and let rest. Boil the pasta: While the chicken cooks, bring a large pot of salted water to a boil. Add the fettuccine and cook until al dente according to package instructions. Reserve 1 to 2 cups of pasta water, then drain the pasta (do not rinse).
Start the sauce: Return the skillet to medium heat and melt the remaining 4 tablespoons of butter. Add the garlic and cook for about 30 seconds to 1 minute, just until fragrant.
Thicken the sauce: Whisk 1-2 tablespoons of flour, ½ teaspoon of salt, and ½ teaspoon of black pepper into the butter. Whisk continuously until it turns a light golden color.
Add the cream: Slowly pour in the heavy cream, whisking constantly until smooth. Bring to a gentle simmer and cook for 3 to 4 minutes, stirring often.
Add the cheese: Reduce the heat to low and stir in the grated Parmigiano Reggiano until melted and smooth. If the sauce feels too thick, add a splash of reserved pasta water until it reaches your desired consistency.
Toss with pasta: Add the drained fettuccine to the skillet and toss until evenly coated in the sauce. Taste and season with additional salt and black pepper, if needed.
Slice and serve: Slice the rested chicken and serve over the pasta or toss it in just before serving. Garnish with chopped parsley, extra Parmesan, black pepper, and a squeeze of lemon juice if desired.