Season chicken pieces with salt, pepper, and 1/4 teaspoon of cayenne pepper.
Mix remaining 1/4 teaspoon of cayenne pepper with flour, sage, and thyme in a bowl. Dredge chicken pieces in the flour mixture, coating each side completely.
Heat oil in a Dutch oven over medium heat. Cook chicken in the hot oil, turning occasionally, until evenly browned, 5 to 6 minutes. Stir in sausage; cook until browned, 5 to 6 minutes. Add onions; cook until softened, 5 to 6 minutes.
Stir chicken stock, garlic, and bay leaves into the Dutch oven; cover and cook until chicken is very tender, about 3 hours. Stir in green onions before serving.