In a large mixing bowl, stir together the flour, baking powder, and salt. In another bowl, whisk together the mashed banana, melted butter, egg, buttermilk, brown sugar, and vanilla extract until smooth. Pour the wet mixture into the dry ingredients and gently fold with a rubber spatula just until everything is incorporated—take care not to overmix the batter. Warm a griddle or large skillet over medium-low heat and lightly grease it with butter. Scoop about ⅓ cup of batter per pancake onto the hot surface, using the back of the cup to spread the batter into even circles. Cook until bubbles begin to appear on the surface; since the batter is thick, the bubbles may be subtle. Flip the pancakes and continue cooking for another couple of minutes, or until they are fully cooked through. Avoid turning the heat too high—slow, gentle cooking helps ensure the centers are done without burning the outsides. Serve immediately with butter, maple syrup, and any desired toppings.