Thinly slice the yellow onions into uniform half-moons to ensure even cooking.
Heat the ghee in a heavy-bottomed pot over medium-low heat and add the onions with a pinch of sea salt.
Cook the onions 'low and slow' for 30-40 minutes, stirring occasionally, until they are soft and reach a rich, dark mahogany color.
Stir in the minced garlic and fresh thyme leaves, cooking for 1 minute until fragrant.
Add the balsamic vinegar to the pot, scraping up any browned bits from the bottom to deglaze.
Pour in the beef bone broth and add the shredded cooked chicken breast, bringing the mixture to a gentle simmer for 10 minutes.
Season the soup with the remaining sea salt and black pepper to taste.
Ladle the soup into an oven-safe crock and top with the shredded Gruyere cheese.
Place the crock under the broiler for 2-3 minutes until the cheese is melted, bubbly, and slightly golden brown.