Heat the oil and butter in a large skillet over medium-high heat; swirl together. Season the chicken breasts lightly with salt and pepper and place in the skillet. Saute for 4 minutes; then turn over the chicken and saute for an additional 3-5 minutes (depending on the thickness of the chicken breasts). Remove breasts from pan and keep warm.
In the same pan, saute the sliced mushrooms and finely diced red onion until golden.
Add 1/2 cup chicken broth, 2 tablespoons balsamic vinegar, 2 sliced cloves of garlic, the dried thyme, and the salt and pepper. Simmer 4 minutes over medium-high heat. Add the breasts and spoon the warm sauce over for a minute.
Remove from heat and stir in the butter until melted into the sauce.