Sprinkle both sides of chicken fillets with salt and pepper. Place 1/3 cup all-purpose flour in a plate and dredge fillets in flour to coat both sides. Shake off excess flour.

Heat oil in a frying pan. Brown chicken on both sides. Transfer fillets to oiled 13x9-inch baking dish.

To make sauce, melt butter in same pan. Add mushrooms and cook until golden brown.

Sprinkle mushrooms with 3 Tbsp flour and let cook, stirring, for 1-2 minutes. Slowly stir in chicken broth, wine and milk. Cook, stirring often, until it begins to thicken and becomes bubbly. Season to taste with salt, pepper and thyme.

Pour mushroom sauce over chicken. Top each fillet with one slice cheese. Cover dish with aluminum foil and bake for 30 minutes in preheated 350 F (175 C).

Remove foil and broil for 1 to 2 minutes to brown the cheese. Serve sprinkled with chopped parsley.