I recommend prepping all ingredients prior to getting this recipe started. Preheat your oven to 375F and position the rack in the top third of the oven.
Cut the chicken breasts in half lengthwise so you've got 6 thinner cutlets. Generously season both sides of each piece with salt & pepper and coat each piece in flour. Shake off any excess flour.
Add the oil and 1 tablespoon of the butter to a skillet over medium-high heat. Let the pan heat up for a few minutes. Cook the chicken in two batches, lightly browning each piece of chicken for about 3-4 minutes per side (don't cook them completely or they'll overcook in the oven). Transfer the chicken to a 9x13 casserole dish as you finish browning each batch. You will likely need to add a splash more olive oil (about a tablespoon or less) to the second batch if the pan is dry.
Add 1 tablespoon of the butter to the same skillet you cooked the chicken in. One it melts, add the mushrooms and cook them until they're seared and their water is released and cooked off (about 5 minutes). Take the skillet off the heat and transfer the mushrooms to the casserole dish (add them around and on top of the chicken as desired).
In the same skillet, over medium heat, add the remaining 2 tablespoons of butter. Once it melts, add the flour and garlic. Cook for about a minute, stirring nearly constantly.
Pour in the sherry and then slowly whisk in the cream until the flour has dissolved and you've got a smooth sauce (and scrape up any brown bits from the bottom of the pan). Cook it for a few minutes (continue whisking) so it thickens up a bit (don't let it get too thick because it'll reduce more in the oven). Add in a bit of salt & pepper to the sauce.
Pour the cream sauce over the chicken and mushrooms.
Top the chicken with the cheese slices.
Cover the casserole with foil and bake for 20 minutes.
Take the foil off and broil for a few minutes until the cheese has browned (watch it carefully).
Let the chicken rest for a few minutes and then serve.