Add butter to large skillet and let it melt. Add the onions. Stir to coat the onions in the butter. Sprinkle with salt and pepper. Cook on med low for 35 minutes stirring occasionally.
Add the garlic, thyme, and rosemary. Turn the heat to high and cook stirring frequently until onions get very dark mahogany color. Stir in 1 cup beef broth. Let cook until most is evaporated. Remove onions with slotted spoon to separate plate and set aside.
Wipe out pan and add some oil. Brown the chicken pieces on both sides. Add the other 1 cup of beef broth to pan. Stir in the onions. Cook until the chicken is no longer pink. Place cheese on top of each piece of chicken and cover with lid and let cheese melt. Serve chicken sitting on top of onions.