For the Alligator Sausage:
Combine the alligator meat, pork fat, black pepper, white pepper, garlic, cayenne pepper, and thyme. Pass through the rough-grind dial of a sausage grinder, then stuff into a natural pork casing. Smoke or steam the sausages according to your preference.
To Assemble and Serve:
Serve the Alligator Sausage on a Toasted Sourdough Bun with relish, cheese, ketchup, mustard, onions, tomato, and jalapeños.