Thinly slice the yellow onions into uniform half-moons.
In a large pot or Dutch oven, melt the ghee over medium-low heat.
Add the onions to the pot with a pinch of sea salt and cook slowly, stirring occasionally, for 25-30 minutes until they are deep golden brown and caramelized.
While onions caramelize, poach or pan-sear the chicken breast until cooked through, then shred into bite-sized pieces.
Once onions are caramelized, stir in the balsamic vinegar and fresh thyme, scraping any browned bits from the bottom of the pan.
Pour in the beef bone broth and add the shredded chicken, black pepper, and remaining sea salt.
Bring the soup to a gentle simmer for 10 minutes to allow the flavors to meld.
Preheat your broiler. Ladle the soup into an oven-safe crock or bowl.
Top the soup with the shredded Gruyere cheese and place under the broiler for 2-3 minutes until the cheese is melted, bubbly, and slightly browned.
Serve immediately while hot.