Prepare the Chicken
Preheat your oven to 375°F (190°C). Season the halved chicken breasts generously with salt and pepper on both sides. Place some flour on a plate and lightly dredge each piece of chicken, shaking off any excess. This thin coating will help the chicken brown beautifully and thicken the sauce later.
Brown the Chicken
Heat the olive oil and 2 tablespoons of butter in a large skillet over medium-high heat. Once hot, add the chicken pieces and brown for 3-4 minutes on each side until golden (they don't need to be cooked through). Transfer the browned chicken to a casserole dish, arranging them in a single layer.
Sauté the Mushrooms
In the same skillet, add another tablespoon of butter. Once melted, add the sliced cremini mushrooms and cook for 4-5 minutes until they release their moisture and begin to brown. Season with a pinch of salt and pepper, then transfer the mushrooms to the casserole dish, distributing them around and over the chicken.
Make the Cream Sauce
Return the skillet to medium heat and add the remaining tablespoon of butter. Once melted, add the minced garlic and cook for 30 seconds until fragrant. Sprinkle in 2 tablespoons of flour and whisk to form a roux, cooking for about 1 minute. Slowly pour in the dry sherry, whisking constantly to prevent lumps. Let it simmer for 1-2 minutes to cook off the alcohol. Gradually whisk in the heavy cream and continue cooking until the sauce thickens enough to coat the back of a spoon, about 3-4 minutes.
Assemble and Bake
Pour the cream sauce evenly over the chicken and mushrooms in the casserole dish. Top each piece of chicken with a slice of Muenster cheese. Cover the dish with foil and bake in the preheated oven for 20 minutes. Then remove the foil and switch to broil for 2-3 minutes, until the cheese is bubbly and beginning to brown. Watch carefully to prevent burning.