Add the flour, and parmesan cheese to a shallow bowl and whisk to combine.
Split large chicken breasts in half (you want 4-6 ounce portions)
Lightly pound them with a meat mallet so that they are flattened out.
Dredge chicken in the flour mixture, coating both sides.
Add the 3 tablespoons of olive oil and the butter to a large skillet over medium-high heat. Swirl the pan to mix the butter and oil
When the oil is hot, add the floured chicken breasts to the pan and cook until golden brown.
Flip the chicken over to cook the other side (3-5 minutes per side, depending on the thickness of the breasts).
Remove the chicken from the pan and place on a plate (tent with foil to keep it warm).
Reduce the heat to medium, and add the chopped onion to the pan and cook for 3-4 minutes until softened. Don't let the onion burn.
Smash 6 cloves of garlic with the side of a large knife.
Add the smashed garlic cloves and whole garlic cloves to the pan and cook until fragrant over medium heat for 2-3 minutes. This begins the caramelization process.*Add more butter or oil if necessary. Add the chicken broth to the pan, scraping all of the tasty bits off the bottom of the pan. *If you'd like to add a splash of white wine to the sauce, do it before adding the chicken stock to the pan. Simmer on low until the broth has reduced by half (4-5 minutes).*This gives the garlic time to fully cook and release the sweet garlicy flavor, instead of a bitter hot flavor. Reduce the heat to medium-low and add the heavy cream
Bring the sauce to a gentle simmer and cook for 2-3 minutes. Stirring as needed.
Add the parmesan cheese to the sauce and continue cooking for 2-3 minutes until the cheese has melted into the sauce. Stirring as needed.
Add the chicken back into the pan.
Let it simmer in the sauce for 2-3 minutes, until the sauce has thickened and the chicken has reached an internal temperature of at least 165 degrees F.
Taste the sauce and season with salt and black pepper as needed.
Serve over mashed potatoes, rice or pasta.
Garnish with chopped Italian parsley.