Set the oven to 375°F (190°C) and position a rack in the upper third.

Slice each chicken breast in half lengthwise to yield six thin cutlets. Season both sides generously with kosher salt and freshly ground black pepper. Dredge the chicken pieces in ⅓ cup of flour, ensuring a light, even coating. Shake off any excess.

In a large skillet over medium-high heat, heat the olive oil and 1 tablespoon of the butter. Sear the chicken cutlets in batches, cooking for approximately 3–4 minutes per side, until golden but not fully cooked through. Transfer the seared chicken to a 9×13-inch casserole dish.

In the same skillet, melt an additional tablespoon of butter over medium heat. Add the sliced mushrooms and cook, stirring occasionally, until the mushrooms are golden brown and their moisture has evaporated—about 5 minutes. Distribute the mushrooms over and around the chicken in the casserole dish.

Return the skillet to medium heat. Melt the remaining 2 tablespoons of butter, then add the minced garlic and cook for approximately 30 seconds until fragrant. Stir in the remaining 2 tablespoons of flour and cook for 1 minute to form a roux.

Gradually pour in the dry sherry, whisking continuously to deglaze the pan and incorporate the roux. Slowly whisk in the heavy cream until the sauce is smooth. Add the red pepper flakes if using. Simmer for 2–3 minutes until slightly thickened. Taste and adjust seasoning with additional salt and pepper as needed.

Evenly pour the cream sauce over the chicken and mushrooms in the casserole dish. Place one slice of Swiss cheese over each chicken cutlet.

Cover the dish with foil and bake for 20 minutes. Remove the foil and broil for 2–3 minutes, watching closely, until the cheese is bubbly and lightly browned.

Allow the dish to rest for 5 minutes before garnishing with freshly chopped parsley. Serve warm.