Thinly slice the yellow onion into half-moons.
Heat the ghee in a heavy-bottomed pot over medium-low heat.
Add the onions to the pot and cook slowly, stirring occasionally, for 25 to 30 minutes until they are deep golden brown and jammy.
Stir in the balsamic vinegar to deglaze the pot, scraping up any browned bits from the bottom.
Pour in the beef bone broth and add the fresh thyme sprig, sea salt, and black pepper.
Add the shredded cooked chicken breast to the pot and simmer for 10 minutes to allow the flavors to meld.
Preheat your oven broiler to high.
Ladle the soup into an oven-safe ramekin or deep bowl.
Top the soup with the shredded Gruyere cheese in an even layer.
Place the bowl under the broiler for 2 to 3 minutes until the cheese is melted, bubbling, and slightly browned.