Preheat the oven to 350°F.
In a large skillet (I use an 11 or 12-inch skillet), melt 1 tablespoon of the butter over medium heat. Add the sliced mushrooms and season lightly with salt and pepper. Sauté, stirring occasionally, until tender and any liquid that has been released from the mushrooms has evaporated. Remove the mushrooms from the pan to a bowl or plate; set aside. Remove the pan from the heat.
Place each chicken breast between two pieces of plastic wrap and gently pound to an even thickness with a meat mallet. Note: We aren't looking for thin breasts for this recipe. Focus on pounding the thicker end of each breast just until the breast evenly thick, so it will cook evenly. If using larger breasts that you cut in half lengthwise, if they are already even in thickness after cutting, you don't need to pound them.
Place the flour on a dish or in a shallow bowl. Season each piece of chicken with salt and pepper and lightly dredge in the flour, shaking off any excess.
Return the skillet to the stove and heat the remaining 2 tablespoons of butter and the olive oil over medium-high heat. Add the chicken and cook just until golden brown on both sides (the chicken should not be cooked through at this point; it will finish cooking in the oven).
Remove the chicken from the pan and place in a single layer in a 9x13-inch baking dish.
Add the chicken broth or stock, sherry, and garlic to the pan. Using a wooden spoon, scrape up any browned bits from the bottom of the pan. Bring to a boil, then reduce the heat and simmer, stirring occasionally, for 2 to 3 minutes to reduce the sauce slightly.
Combine the cold water and cornstarch in a small bowl; stir until smooth. While stirring (I like to use a whisk here), add the mixture to the sauce. Continue simmering and stirring for another minute or two just until the sauce has thickened slightly (it should not be a super thick sauce; if you feel it is too thick, add a little more broth). Taste and season with salt and pepper if needed.
Top the chicken breasts with the cooked mushrooms.
Pour the sauce over the chicken and mushrooms.
Top each piece of chicken with a slice of cheese (I cut each slice of cheese in half to get better coverage).
Bake for 15 minutes or until the chicken is cooked through (internal temperature of at least 165°F) and the cheese is melted.
Garnish with chopped fresh parsley, if desired, and serve.