Preheat oven to 375 degrees Fahrenheit and spray a 9x13-inch baking dish with cooking spray.
Season the chicken with salt and pepper. Reserve 3 tablespoons of flour and coat the chicken with the remaining flour.
Heat 3 tablespoons of the butter in a skillet over medium-high heat. Cook the breaded chicken in the butter for 3-4 minutes on each side, just to brown them, don’t cook the chicken all the way through.
Place the browned chicken into the prepared baking dish. Add the remaining 3 tablespoons of butter to the skillet.
Add the onion and mushrooms to the skillet with the butter. Cook the onion and mushroom until they are tender, about 4-5 minutes.
Add the garlic and cook for just one minute, until the garlic is fragrant. Whisk the reserved 3 tablespoons of flour into the butter mixture in the skillet to create a paste.
Slowly whisk the heavy cream, chicken broth, and red wine vinegar into the paste mixture to create a sauce, and season with salt and pepper to taste.
Pour the sauce over the chicken in the baking dish. Place the slices of Muenster cheese over each chicken cutlet, cover the dish with foil, and bake for 20 minutes.
Remove the foil from the dish, set the oven to broil, and broil for 2-3 minutes until the cheese is melted and bubbly.
Remove the dish from the oven and let it cool for a few minutes before serving.