In a large pot over medium-low heat, melt the ghee and add the thinly sliced onions.
Cook the onions for 25-30 minutes, stirring occasionally, until they are deep golden brown and caramelized.
While the onions cook, poach or sear the chicken breast until fully cooked through, then shred it into bite-sized pieces.
Stir the minced garlic and fresh thyme into the caramelized onions, cooking for 1 minute until fragrant.
Pour in the balsamic vinegar to deglaze the pot, scraping up any browned bits from the bottom with a wooden spoon.
Add the beef broth, sea salt, and black pepper, then bring the liquid to a gentle simmer for 10 minutes.
Stir in the shredded chicken breast and let the soup heat through for an additional 5 minutes.
Ladle the soup into an oven-safe crock or bowl and place the toasted sourdough slice on top.
Sprinkle the shredded Gruyère cheese over the bread and broil in the oven for 2-3 minutes until the cheese is melted and bubbly.