Heat ghee in a large pot over medium-low heat and add the sliced yellow onions.
Cook the onions for 25-30 minutes, stirring occasionally, until they are deeply caramelized and golden brown.
Stir in the balsamic vinegar, sea salt, black pepper, and fresh thyme leaves.
Pour in the beef bone broth and add the whole chicken breast to the pot.
Simmer for 15-20 minutes until the chicken is cooked through, then remove the chicken and shred it with two forks.
Return the shredded chicken to the pot and simmer for another 5 minutes to allow the flavors to meld.
Preheat your broiler to high.
Ladle the soup into an oven-safe crock and place the sourdough slice on top, followed by the shredded Gruyere cheese.
Broil for 2-3 minutes until the cheese is melted, golden, and bubbling.