In a bowl, whisk together the flour, baking powder, baking soda, and salt until thoroughly combined. In a large bowl, whisk the eggs until well beaten, then whisk in sour cream and vanilla extract. Whisk the flour mixture into the sour cream mixture, and mix in the milk and melted butter to form a thin but smooth batter.
Butter a skillet, and heat over medium heat until the butter is fragrant; pour in enough batter to coat the skillet in a thin layer (pancakes should be very thin). Allow to cook until the bottom is lightly browned, then turn the thin pancake over to brown the other side.