Thinly slice the onion. Place in frying pan with butter and sugar and cook on low hear for about 20 minutes until lightly browned and caramelized.
Meanwhile, combine the bison with the mustard, salt, pepper, onion powder, garlic powder, paprika, parsley and cayenne. Mix gently and form into 4 patties. Indent slightly in the middle of each patty, to help maintain their shape while cooking.
Drizzle grill pan with avocado oil and heat on medium heat for 5 minutes. Cook for 4 minutes per side. After flipping, add cheese to each patty to melt while cooking the other side.
Serve on a roll, topping each patty with some of the caramelized onions and your favorite BBQ sauce.