Slice each chicken breast horizontally to create four thin chicken cutlets.
Season both sides with salt and black pepper, then lightly coat each piece with flour, shaking off any excess.
Heat 1 tablespoon butter in a large skillet over medium-high heat.
Add the chicken and cook for about 2½ minutes per side until golden brown and fully cooked. Transfer the chicken to a plate and cover loosely with foil to keep warm.
In the same skillet, melt the remaining butter over high heat.
Add the sliced mushrooms and cook for about 4 minutes until they become golden brown around the edges.
Stir in the minced garlic along with a pinch of salt and pepper. Cook for another minute until fragrant.
Pour in the white wine (or additional chicken stock) and scrape up any browned bits from the bottom of the skillet. Simmer for about 30 seconds until most of the liquid has evaporated.
Add the chicken stock and cook for another minute until slightly reduced.
Lower the heat to medium and stir in the heavy cream. Simmer gently for about 2 minutes until the sauce begins to thicken.
Add the freshly grated Parmesan cheese and stir until melted and smooth. Taste and adjust the seasoning with additional salt and pepper if needed.
Return the chicken breasts to the skillet and spoon the creamy mushroom sauce over the top. Simmer for about 30 seconds until heated through.
Garnish with chopped chives or fresh parsley before serving.