Begin by preheating the oven to 375°F. Prepare a 9x13-inch baking dish by lightly coating it with nonstick cooking spray.
Season the chicken cutlets generously with salt and pepper on both sides. Reserve 3 tablespoons of flour for later use, then dredge the cutlets in the remaining flour, ensuring they are coated evenly.
Heat 3 tablespoons of butter in a skillet over medium-high heat. Add the floured chicken cutlets to the pan, cooking each side for 3-4 minutes until they develop a light golden brown color. The chicken should not be fully cooked at this stage. Once seared, transfer the cutlets to the prepared baking dish.
In the same skillet, melt the remaining 3 tablespoons of butter. Sauté the minced onion and diced mushrooms, stirring frequently until they soften and release their aroma, about 4-5 minutes.
Add the minced garlic to the skillet and sauté for an additional minute until fragrant. Sprinkle the reserved 3 tablespoons of flour over the mixture, stirring to form a roux-like paste.
Gradually whisk in the heavy cream, chicken broth, and red wine vinegar, combining the ingredients thoroughly. Taste and adjust the seasoning with additional salt and pepper as needed.
Pour the prepared sauce evenly over the chicken cutlets in the baking dish. Place a slice of Muenster cheese on top of each cutlet, ensuring full coverage. Cover the dish with aluminum foil and bake for 20 minutes.
After baking, remove the foil and switch the oven to broil. Broil the dish for 2-3 minutes or until the cheese becomes bubbly and slightly golden.
Allow the dish to rest for a few minutes before serving. Enjoy your Chicken Gloria!