Place a chicken breast into a large Ziploc bag, close the bag almost all the way, then pound the chicken to a thickness to 1/2-inch thick; repeat with remaining breasts.
Mix together salt, pepper, flour, and sage.
Dredge flattened breasts in a mixture; shake off excess flour.
Melt together the olive oil and butter in a large wide sauté pan or large electric griddle (one with a cover) and brown the chicken over medium-high heat for 3 minutes on one side.
Add the garlic and shallot then turn the chicken over and sprinkle with the mushrooms.
Saute for an additional 3 minutes, stirring the mushrooms so that they will cook evenly.
(You may need to cook the chicken one or two at a time if you don't have a large pan; if so, cook them all then add them all back into the pan for the final steps.) Stir in the balsamic vinegar, chicken broth, and wine.
Cover and simmer over medium-low heat for 10 minutes, basting with the sauce and turning the chicken once or twice.
Season with salt and pepper to taste and serve.
Makes 4 servings.
You can also cook the breasts without flattening them first.