Place banana in a mixing bowl and mash into a smooth paste with a potato masher. Add egg and sugar and whisk thoroughly until sugar is dissolved and mixture is smooth and creamy, about 2 minutes.

Add milk, vanilla extract, 1/4 cup melted butter, walnuts, and flour mixture. Whisk until just combined.

Let batter rest for 10 minutes.

Melt 1 tablespoon butter in a large, nonstick pan or skillet over medium-high heat. Once the pan is hot, transfer in about 1/3 cup of the batter per pancake. Reduce heat to medium and cook until edges start to look dry and small air bubbles pop up through the surface, about 3 minutes.

Flip and cook until golden brown, about 3 minutes more. Serve immediately.