Blend oats in a blender until the texture resembles coarse flour.
Transfer oats to a bowl. Add both flours, brown sugar, milk powder, baking powder, baking soda, and salt; whisk until well combined.
Whisk together milk, egg, 2 tablespoons vegetable oil, and vanilla in a separate bowl; stir in mashed banana. Pour egg mixture into flour mixture; stir just until moistened, then let batter stand for 5 minutes.
Lightly oil a griddle with 1 teaspoon oil and heat over medium-high heat.
Working in batches, drop batter by large spoonfuls onto the hot griddle; cook until bubbles form and the edges are dry, about 2 minutes. Flip, and cook until browned on the other side. Repeat with remaining batter.