Season the chicken: Pat the chicken dry and season both sides with 1/2 teaspoon salt, the pepper, and Italian seasoning.
Sear: Heat the olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Sear the chicken 5-6 minutes per side until deeply golden and cooked through (165°F internal). Transfer to a plate and tent with foil.
Sauté the garlic: Reduce heat to medium. Add the remaining 2 tablespoons butter and the garlic; cook 30-60 seconds until fragrant.
Make a roux: Sprinkle the flour over the garlic and stir 1 minute — this thickens the sauce so it coats beautifully.
Build the sauce: Slowly whisk in the chicken broth, scraping up the browned bits from the pan. Whisk in the heavy cream and simmer 3-4 minutes until slightly thickened.
Add the parmesan: Reduce heat to low and stir in the parmesan a handful at a time until melted and smooth. Stir in the lemon juice if using, and season with the remaining 1/2 teaspoon salt to taste.
Finish: Return the chicken (and any juices) to the skillet. Spoon sauce over the top and simmer 2-3 minutes until heated through.
Serve: Garnish with parsley and extra parmesan. Serve over pasta, mashed potatoes, or rice.