Melt 2 tablespoons unsalted butter and let cool slightly, 2–3 minutes.
In a medium bowl, whisk together the all-purpose flour, granulated sugar, baking powder, and salt until evenly combined.
In a separate large bowl, mash the banana until mostly smooth. Whisk in the beaten egg, milk, vanilla extract, and the melted butter until well combined.
Add the dry ingredients to the wet and stir just until no dry flour remains; a few small lumps are fine. Let the batter rest 5 minutes to hydrate.
Heat a nonstick skillet or griddle over medium heat. Add a thin film of the remaining unsalted butter for cooking and swirl to coat.
Ladle batter in 1/4-cup portions for each pancake. Cook until the edges look set and bubbles form across the surface, 2–3 minutes. Flip and cook until golden brown and cooked through, 1–2 minutes more.
Adjust heat as needed to maintain gentle browning. Re-butter the pan between batches and repeat with remaining batter.