Gently mix chicken, squeezed apple, sage, fennel, salt, and pepper. Shape four equal patties about 4 1/2 inches wide and 1/2 inch thick.
Heat oil in a large nonstick skillet over medium. Cook uncrowded patties 5 to 6 minutes per side until browned and centers reach 165°F; lower heat if darkening too quickly.
Meanwhile, mix softened butter and maple syrup until smooth. Broil only cut bagel faces 1 to 2 minutes.
Spread maple butter evenly over both warm cut faces. Place one cooked sausage patty on each bottom.
Close; rest 1 minute and halve. Keep patties intact to retain juices rather than soak bread.