Thinly slice the yellow onions into uniform half-moons.
In a large pot, melt the ghee over medium-low heat and add the onions with the sea salt.
Sauté the onions for 25-30 minutes, stirring occasionally, until they are deep golden brown and caramelized.
Stir in the balsamic vinegar and fresh thyme, scraping the bottom of the pot to release the flavorful browned bits.
Pour in the beef broth and add the cooked shredded chicken breast, then bring the soup to a gentle simmer for 10 minutes.
Season with black pepper and adjust salt if necessary.
Toast the sourdough slice until crisp and firm.
Ladle the soup into an oven-safe crock, place the toast on top, and cover with the shredded Gruyere cheese.
Broil for 2-3 minutes until the cheese is melted, bubbly, and golden brown.