Season the chicken breasts on both sides with a sprinkle of kosher salt and pepper.
Spread the flour out on a large plate and sprinkle with salt and pepper. One at a time, dip the chicken breasts in the flour, turning to coat. Shake lightly to remove any excess, then set the chicken breast aside on a clean plate. Repeat with all 4 chicken breasts.
Add the oil to a large nonstick skillet and warm over medium-high heat. Add the chicken breasts to the pan, and cook on one side for about 6 minutes, until nicely golden brown.
Turn the chicken breasts over and add the garlic cloves and mushrooms to the pan. Cook for another 3 or 4 minutes, shaking the pan occasionally to redistribute the mushrooms and garlic while they cook.
Add the balsamic vinegar, chicken stock, bay leaf, and thyme to the pan. Reduce the heat to a simmer, cover the pan, and continue to cook for 10 minutes, turning the chicken breasts halfway through.
Uncover the pan and remove the chicken breasts to a plate. You can lightly tent them with foil to keep them warm.
Let the balsamic-mushroom sauce continue to cook, uncovered, for another 5 minutes or so, until reduced to a sauce consistency. Stir in the butter and cook for another 2 minutes or so until melted and combined with the rest of the sauce.
Remove the bay leaf, then plate each chicken breast and spoon the mushroom sauce over the top. Enjoy!