Season chicken cutlets on both sides generously with salt and pepper.
Heat olive oil in a large cast iron skillet over medium high heat. Add chicken and cook for 3-4 minutes per side, until golden brown, cooked through and juices run clear. Transfer to a plate on the side and cover with foil.
Reduce heat to medium and add onions to the same skillet along with another 1-2 T oil. Cook for 5-6 minutes, then add 2 T butter, garlic, and fresh thyme. Cook another 2-3 minutes then add balsamic vinegar and stir to combine. Continue cooking another 5-6 minutes, stirring frequently, until onions are a deep golden brown (but not burned!).
Reduce heat to medium low. Pour in beef broth and half and half and stir to combine. Add chicken back to skillet and spoon onions and juices over the tops of the chicken. Add gruyere cheese to the tops of the chicken and simmer on low until cheese is melted.