Preheat the Oven – Set oven to 375°F and position a rack in the top third.
Prepare the Chicken – Slice each chicken breast in half lengthwise, season with salt and pepper, and dredge in 1/3 cup flour, shaking off excess.
Sear the Chicken – In a skillet over medium-high heat, warm olive oil and 1 tablespoon butter. Sear chicken for 3–4 minutes per side, then transfer to a 9×13-inch casserole dish.
Cook the Mushrooms – Melt 1 tablespoon butter in the skillet, sauté mushrooms until golden (about 5 minutes), and add them to the dish.
Make the Sauce – Melt 2 tablespoons butter in the skillet, sauté garlic for 30 seconds, then whisk in remaining 2 tablespoons flour. Cook for 1 minute.
Add Liquids – Pour in sherry, whisking to deglaze the pan, then gradually add heavy cream, stirring until smooth. Simmer for 2–3 minutes. Add red pepper flakes (if using) and adjust seasoning.
Assemble – Pour sauce over the chicken and mushrooms, then top each piece with Swiss cheese.
Bake & Broil – Cover with foil and bake for 20 minutes. Remove foil and broil for 2–3 minutes until the cheese is bubbly and golden.
Finish & Serve – Let rest for 5 minutes, garnish with parsley, and enjoy!