Melt the ghee in a medium pot over medium-low heat.
Add the sliced yellow onion and cook for 25-30 minutes, stirring occasionally, until deeply caramelized and golden brown.
Stir in the minced garlic, fresh thyme, sea salt, and black pepper, cooking for 2 minutes until fragrant.
Pour in the beef bone broth and bring to a gentle simmer for 10 minutes to meld the flavors.
Stir in the pre-cooked shredded chicken breast until heated through.
While the soup simmers, toast the sourdough bread and rub it with the raw garlic clove.
Ladle the soup into an oven-safe crock, place the toasted bread on top, and sprinkle with the shredded Gruyere cheese.
Place under the broiler for 2-3 minutes until the cheese is melted, bubbly, and slightly browned.