Gently mix ground chicken, squeezed grated apple, sage, fennel, salt and pepper in a bowl. Divide into four equal portions and shape into patties about 4 1/2 inches wide and 1/2 inch thick.
Heat oil in a large nonstick skillet over medium heat. Add patties without crowding and cook for 5 to 6 minutes per side until browned and 165°F in the center, lowering heat if the outsides darken too quickly.
Meanwhile, stir softened butter and maple syrup together until smooth. Toast only the cut bagel faces under the broiler for 1 to 2 minutes.
Spread the maple butter evenly over both cut faces of the warm bagels. Put one cooked chicken sausage patty on each bottom.
Close with the tops, rest for 1 minute and cut in half. Keep the patties intact so their juices stay in the meat rather than soaking straight into the bread.