1.Preheat the grill to medium-high.
2.Using a food processor, process the andouille until finely chopped.
3.Transfer the andouille to a large bowl. Add chicken, salt, pepper, garlic powder, onion powder and cayenne. Mix gently, being careful not to overwork the mixture.
4.Form the mixture into 12 patties, and place them on the grill. Cook until the internal temperature registers 170 degrees F, about 5 minutes per side. During the last minute of grilling, add the cheese to the tops of the burgers, and toast the rolls (cut-side down, about 30 seconds).
5.Remove the burgers and buns from the grill. Garnish with your choice of toppings and condiments. Serve immediately.