Melt the butter and set it aside to cool for a minute.
In a large bowl, whisk together the flour, sugar, baking powder, baking soda, and salt.
In a separate bowl, whisk the egg and buttermilk until smooth, then whisk in the melted butter and vanilla.
Pour the wet ingredients into the dry ingredients and stir gently until just combined. A few lumps are fine.
Set a 10-inch skillet over medium heat and brush it lightly with ½ tablespoon of butter or oil.
Pour ½ to ¾ cup of batter into the skillet. Cook until the edges look dry and bubbles form and pop on top, about 2 to 3 minutes.
Flip and cook until the second side is golden, about 1 to 2 minutes more.
Repeat with the remaining batter, adding a little more butter or oil to the skillet as needed.
Serve warm.
Notes
Don't overmix the batter: Stop stirring the moment the dry flour disappears. Overmixing works the gluten in the flour and makes buttermilk pancakes dense and flat.Let the first pancake set your heat: Cook one pancake first, then look at it. If it browned too fast or too slowly, adjust the heat before you cook the rest. Medium heat usually gives an evenly golden pancake.Use a 10-inch skillet: The batter makes large pancakes, about ½ to ¾ cup each, so cook them one at a time in a 10-inch skillet to give each one room to spread.Flip each pancake only once: Once you flip, leave it alone. Flipping again or pressing it with the spatula pushes out the air and flattens it.Keep the finished pancakes warm: Since you cook them one at a time, hold the finished pancakes on a plate in a 200°F oven so all three are warm when you sit down.