Grilled Longganisa

Name: Grilled Longganisa
Ingredients: 2 to 4 hog casings, plus more if needed (some may tear); 4 pounds boneless pork shoulder, cut into 3/4-inch cubes; 2 pounds skinless pork belly or pork fatback, frozen until firm, then cut into 3/4-inch cubes; 1 head garlic, minced; 1 1/2 cups packed light brown sugar; Kosher salt and freshly ground pepper; 1/2 cup vegetable oil, plus more for the stuffer attachment and the grill; 1/2 cup annatto seeds; 1/4 cup soy sauce; 1/4 cup coconut vinegar or distilled white vinegar