Process the rolled oats in a food processor until the texture resembles coarse flour. In the medium bowl, whisk together the oat flour, whole wheat flour, all-purpose flour, brown sugar, baking powder, baking soda, and salt in a bowl. Set aside.
In the small bowl, whisk together the egg, buttermilk, and melted butter. Stir in the mashed banana. Pour the egg mixture into the flour mixture and stir just until moistened. Let the batter stand for 5 minutes.
Warm a lightly oiled griddle over medium-high heat. Ladle the batter onto the hot griddle and cook until bubbles form and the edges turn dry, about 2 minutes. Flip and cook until browned on the other side. Keep warm on a covered plate while cooking the rest. Repeat with remaining batter.