Add ghee to a large stockpot over medium heat. Once melted, add onions, carrots, and celery. Allow to saute for approximately 5 minutes.
Add fish bones and fish heads on top of vegetables and add white vermouth. Cover stockpot and allow bones to "sweat" for approximately 10 minutes.
Remove lid and add enough hot water to cover bones. Bring water up to a high simmer and skim any foam that comes up to the top.
Turn heat to low and allow bone broth to simmer, uncovered, for 30 minutes.
After 30 minutes, remove the fish bones, fish heads, and vegetables from the stockpot and then strain the broth and decant into a clean container.
Broth will stay fresh for 2-3 days in the refrigerator and 2-3 months in the freezer.