Preheat the oven to 350°F (175°C) and prepare a 13 x 9-inch baking dish.
Prepare aromatics: mince 4 garlic cloves, chop ½ medium yellow onion, and slice 6 oz mushrooms.
In a small cup, whisk ½ cup whole milk with 2 teaspoons cornstarch until smooth; set aside.
Place ¼ cup all-purpose flour in a shallow dish. Slice each of the 2 large chicken breasts in half lengthwise to make 4 cutlets; season both sides with ½ teaspoon salt and ¼ teaspoon black pepper and dredge each piece in the flour, shaking off excess.
In a large skillet over medium-high heat, melt 1 tablespoon unsalted butter with 1 tablespoon olive oil. Brown the floured chicken cutlets 2–3 minutes per side until golden; transfer to a plate and set aside.
Add the chopped onion and minced garlic to the skillet and sauté until the onion is almost translucent, about 2–3 minutes; add a bit more butter if the pan is dry.
Add the sliced mushrooms and ½ teaspoon dried thyme to the skillet and cook, stirring often, for 4–5 minutes until the mushrooms soften.
Pour in ¾ cup chicken broth and deglaze the pan, scraping up any browned bits from the bottom with a wooden spoon.
Stir the milk–cornstarch mixture, then add it to the skillet. Simmer the sauce until it thickens slightly, about 1–2 minutes.
Arrange the browned chicken cutlets in the prepared baking dish, then pour the mushroom sauce over the chicken, using tongs to move pieces so sauce coats the bottom of the dish.
Top the chicken and sauce with 6 to 8 slices of Muenster cheese.
Bake for 10–15 minutes, until the cheese is melted and the sauce is bubbling around the edges. Remove from oven and let rest a few minutes before serving.