Season the chicken breasts with salt and pepper on both sides. Lightly dredge each chicken breast in all-purpose flour, shaking off any excess.
In a large skillet, heat 2 tablespoons of butter and 2 tablespoons of olive oil over medium-high heat. Add the chicken breasts to the skillet and cook for about 4-5 minutes per side, or until golden brown and cooked through. Remove the chicken from the skillet and set aside.
In the same skillet, add the sliced mushrooms. Sauté for about 5-7 minutes, or until the mushrooms are browned and tender. Remove the mushrooms from the skillet and set aside with the chicken.
In the same skillet, melt the remaining 2 tablespoons of butter over medium heat. Sprinkle 1 tablespoon of flour over the melted butter and whisk continuously for about 1-2 minutes, until the flour is lightly browned and forms a roux. Gradually whisk in 1 cup of white wine, scraping up any browned bits from the bottom of the pan. Let the wine simmer for 2-3 minutes to reduce slightly.
Stir in 1 cup of chicken broth and bring to a simmer. Let the sauce cook for about 5 minutes, or until it thickens to your desired consistency. If using, stir in fresh thyme leaves. Stir in the fresh thyme leaves.
Return the cooked chicken breasts to the skillet, spooning some of the thickened sauce over them. Top the chicken breasts with cooked mushrooms and place a slice of prosciutto on each, followed by about ¼ cup of shredded Fontina cheese.
Preheat your oven's broiler. Place the skillet under the broiler for 2-3 minutes, or until the cheese is melted and bubbly.
Serve the Chicken Valdostana hot, drizzled with the white wine sauce and garnished with chopped fresh parsley.