In a large ovenproof skillet, heat the butter and 1 tbsp of olive oil over medium heat until hot. Add in the onions and season with salt and pepper.
Saute for 5 minutes. Reduce heat to medium-low and stir the onions frequently until caramelized, about 20 minutes.
Add in the balsamic and stir for 1 minute.
Remove the onions to a large bowl and reserve.
Increase the heat back to medium heat and add remaining olive oil. Add the chicken to the pan and cook for 7-8 minutes. Flip and cook for an additional 5 minutes or until chicken registers 165 with a meat thermometer.
Preheat broiler.
Remove the chicken to a plate and reserve.
Add the flour to the pan, and stir for 1 minute.
Pour in the chicken broth and stir until thickened and the consistency of a gravy.
Add the onions an chicken back to the pan.
Top each breast with a baguette slice, and top with cheese.
Broil for 3-4 minutes or until cheese is melted and browned.
Top with thyme.