Slice the chicken breasts in half horizontally to make 4 cutlets.
Put each cutlet between two sheets of plastic wrap and lightly pound them until they are about 1/4 inch thick. Season with 1/2 teaspoon salt and a pinch of black pepper.
Place an 18-inch long length of wax paper on the counter. Put the flour in a shallow bowl and lightly dredge the chicken pieces in the flour, shaking off any excess. Put the chicken on the wax paper, reserve the 1 teaspoon remaining flour to use later.
Heat a large nonstick skillet over medium-high heat. Add the 1/2 Tbsp butter and tsp of the olive oil to the pan and swirl the pan until the butter has melted. Add the chicken and cook until slightly golden on both sides. Transfer to baking dish and place in the oven to keep warm.
Add the remaining butter and olive oil to the pan. Cook the garlic and shallots until soft and golden. Next, add the mushrooms, season with salt and pepper and cook until soft. Sprinkle the reserve flour and cook for 30 seconds. Add the wine, broth and parsley. Cook scraping up and browned bits from the bottom of the pan, about 2 minutes.
Return the chicken to the skillet with the mushrooms, reduce heat and simmer in the sauce for about 5 minutes.
Serve with mushrooms on top, with a salad with rice.