Heat the butter in a saucepan over medium heat. Add the onions to the pan and cook for about an hour stirring occasionally.
Stir in the flour, garlic, and thyme. Continue cooking for another minute.
Half of the broth should be added.
Heat the oil in a large skillet over medium heat. Season the chicken with salt. Add the chicken to the pan and brown the skin side-by-side for about 2 minutes.
The chicken broth that was left over from the previous cooking time should be added to the saucepan. Deglaze the pan and add the mustard and balsamic vinegar. Reduce the heat to low and simmer for about 5 minutes.
Bake the onions in an oven-safe pan for 20 minutes. Top with the chicken broth and chicken. Cover the pan with foil or a lid and bake in the oven at 350F/180C for another 20 minutes.
Broil the cheese until it is melted.