Heat griddle, not too high. Combine flour, baking powder, baking soda, salt, and sugar in a medium bowl and whisk or stir with a fork. In a separate bowl, whisk together the beaten eggs, buttermilk, and 4 tablespoons of the melted butter. Add the wet ingredients to the dry ingredients and very gently fold them together. Do not overmix; the batter should have small to medium lumps.
Heat oven to 200 degrees. Test griddle by sprinkling a few drops of water on it. If water bounces and spatters off griddle, it is hot enough. Using a pastry brush, brush remaining 1/2 teaspoon of butter onto griddle. Wipe off excess.
For each pancake, pour about 1/2 cup of batter. When pancakes have bubbles on top and are slightly dry around edges, about 2 1/2 minutes, flip them over. Cook until golden on bottom, about 1 minute.