Heat the extra virgin olive oil in a medium pot over medium-low heat.
Add the thinly sliced yellow onions and sea salt, cooking slowly for 20-25 minutes while stirring occasionally until the onions are deeply caramelized and golden brown.
Stir in the minced garlic and dried thyme, cooking for 1 minute until the aromatics are fragrant.
Pour in the balsamic vinegar to deglaze the bottom of the pot, using a wooden spoon to scrape up any flavorful browned bits.
Add the beef bone broth and the whole chicken breast to the pot, then bring the liquid to a gentle simmer.
Cover and cook for 12-15 minutes until the chicken is fully cooked through, then remove the chicken and shred it with two forks.
Return the shredded chicken to the pot and season the soup with black pepper.
While the soup finishes simmering, lightly toast the sourdough bread slice until crisp.
Ladle the soup into an oven-safe crock, place the toasted bread on top, and sprinkle with the shredded Gruyere cheese.
Place the crock under the oven broiler for 2-3 minutes until the cheese is melted, golden, and bubbly.