Dredge the chicken: Season the flour with salt and pepper, then coat each chicken cutlet, shaking off the excess.
Sear the meat: Heat the oil and 1 tbsp of butter in a large skillet. Fry the chicken for 3-5 minutes per side until golden-brown and cooked through. Remove and set aside.
Sauté the aromatics: In the same skillet, melt the remaining butter. Add the minced garlic and cook for 1 minute until fragrant and golden.
Simmer the sauce: Pour in the chicken broth and heavy cream. Bring to a gentle simmer, scraping up the flavorful brown bits from the bottom of the pan.
Add the cheese: Stir in the Parmesan cheese and Italian seasoning. Whisk constantly until the cheese is melted and the sauce has thickened.
Combine: Return the chicken to the skillet, spooning the creamy sauce over each piece.
Garnish and serve: Top with a flurry of fresh parsley and serve immediately while the sauce is at its creamiest.