Heat the ghee in a medium pot over medium-low heat.
Add the thinly sliced yellow onion and cook slowly, stirring occasionally, for about 20 to 25 minutes until the onions are soft and deeply caramelized.
Stir in the sea salt, black pepper, and dried thyme to coat the onions.
Add the balsamic vinegar to the pot, scraping up any browned bits from the bottom.
Pour in the beef bone broth and add the shredded cooked chicken breast, then bring the soup to a gentle simmer for 10 minutes.
While the soup simmers, lightly toast the sourdough bread slice.
Ladle the soup into an oven-safe bowl, place the toasted bread on top, and sprinkle with the shredded Gruyere cheese.
Place the bowl under the broiler for 2 to 3 minutes until the cheese is melted, golden, and bubbling.