Slice each chicken breast in half horizontally to create 4 thinner cutlets. Pat them dry with paper towels. Season both sides with salt and pepper, then lightly coat with flour, shaking off any excess.
In a large skillet, heat 1 tablespoon of butter over medium-high heat. Add the chicken and sear for about 4–5 minutes per side, or until golden brown and cooked through (internal temp should be 165°F/74°C). Remove from the skillet and set aside.
In the same skillet, melt 2 tablespoons of butter. Add the sliced mushrooms and cook for 4-5 minutes, stirring occasionally, until browned and tender. Add the minced garlic and cook for 1 more minute until fragrant.
Pour in the white wine and let it simmer for 2–3 minutes, scraping up any browned bits from the bottom of the pan.
Add the chicken stock and heavy cream (or half-and-half). Stir and bring the mixture to a gentle simmer. Let it cook for 3–4 minutes, until slightly thickened. Then stir in the Parmesan cheese until melted and smooth.
Return the cooked chicken to the skillet, nestling it into the sauce. Let it simmer for another 2–3 minutes so the chicken absorbs the flavors.
Spoon the creamy mushroom sauce over the chicken, garnish with fresh herbs and serve warm. It's great with mashed potatoes, pasta, or steamed green beans.