Warm th soil and butter in a large stainless steel skillet over medium-high heat. Add the onions and toss to coat. Cook for 4 minutes, stirring occasionally, then add the garlic and continue cooking for 1 minute, until the onions are tender an lightly browned.

While the onions are cooking, prep the mushrooms and chicken. Place the chicken pieces in a medium- size bowl. Sprinkle with flour, 1/2 teaspoon of the salt and 1/4 teaspoon of pepper. Toss to coat. Alternately, place the ingredients in a large zip-top bag, seal and shake to coat.

Push the onions to the side of the skillet. Add the chicken in a single layer, leaving the onions at the side or on top of the chicken as it cooks. Cook the chicken for 3 minutes on each side, using a metal spatula to lift and turn the pieces. The chicken shouldn't be fully cooked yet, but it will be white or lightly browned all over. Transfer the chicken and onions to a plate.

Place the skillet back on the stove over medium-high heat. Add the chicken broth, balsamic vinegar, mushrooms, remaining 1/4 teaspoon of salt and remaining 1/4 teaspoon pepper to the pan and bring to a simmer. Use a metal spatula to drape up all the browned bits from the bottom of the pan as the mushrooms cook for 2 to 3 minutes. Add the cream, stir and transfer the chicken and onions back to the skillet. Simmer for about 3 minutes, until the chicken is cooked through and the sauce thickens slightly. Remove from the heat and store in the fresh thyme.