In a large mixing bowl, whisk together the flour, sugar, baking powder and salt until evenly combined.
In a separate bowl or large measuring cup, whisk the milk, egg, melted butter and vanilla until smooth.
Pour the wet ingredients into the dry ingredients and stir gently with a whisk or spatula just until no dry flour remains. The batter should be thick and lumpy; do not try to smooth it out.
Let the batter rest for 5 minutes while you heat a non-stick skillet or griddle over medium-low heat. It is ready when a few drops of water sizzle and skitter across the surface.
Lightly grease the pan with a little butter. Pour 1/4 cup of batter per pancake into the pan, leaving space between them.
Cook for 2 to 3 minutes, until bubbles appear across the surface, the edges look set and dry, and the underside is golden brown.
Flip each pancake once and cook for another 1 to 2 minutes, until the second side is golden and the center springs back when lightly pressed.
Transfer to a plate and repeat with the remaining batter, adding a little more butter to the pan as needed. Serve warm. The batter makes about 10 to 12 pancakes.