Thinly slice the yellow onion into uniform half-moons.
In a heavy-bottomed pot, melt the ghee over medium-low heat.
Add the onions and cook slowly, stirring occasionally, for 20-25 minutes until they are deep brown and caramelized.
Stir in the balsamic vinegar and fresh thyme, using a wooden spoon to scrape up any browned bits from the bottom of the pan.
Pour in the beef bone broth and add the shredded cooked chicken breast.
Bring the soup to a gentle simmer and cook for 10 minutes to allow the flavors to meld.
Season the soup with sea salt and black pepper to taste.
While the soup simmers, toast the sourdough slice in a toaster or under the broiler until crisp.
Ladle the soup into an oven-safe crock or bowl, place the toasted bread on top, and sprinkle with shredded Gruyere cheese.
Place the bowl under the broiler for 2-3 minutes until the cheese is melted, bubbly, and slightly browned.