Heat the ghee in a large heavy-bottomed pot over medium-low heat.
Add the thinly sliced yellow onions and cook slowly for 25 to 30 minutes, stirring occasionally, until they are deep golden brown and caramelized.
Stir in the minced garlic and fresh thyme, cooking for 1 minute until the aromatics are fragrant.
Pour in the balsamic vinegar to deglaze the pot, using a wooden spoon to scrape up all the flavorful browned bits from the bottom.
Add the beef bone broth and the cooked, shredded chicken breast to the pot.
Season the soup with sea salt and black pepper, then bring to a gentle simmer for 10 minutes to meld the flavors.
While the soup simmers, lightly toast the half-slice of sourdough bread until crisp.
Ladle the hot soup into an oven-safe crock or bowl, place the toasted sourdough on top, and sprinkle evenly with the shredded Gruyere cheese.
Place the bowl under a broiler for 2 to 3 minutes until the cheese is melted, golden brown, and bubbly.