Mix the flour, sugar, baking powder, and salt in a large bowl.
In another bowl, whisk the milk, butter, eggs, and vanilla.
Pour the mixture into the large bowl and whisk just enough to combine thoroughly. (Don't over-whisk, or the pancake will be tough. It's ok if there're some lumps left in the batter. When you cook it it'll smooth out.).
Heat the griddle until a few drops of water tossed onto the surface runs around frantically.
Melt a pad of butter onto the griddle and, using a paper towel, rub the butter around until it evenly coats the pan and lubricates it.
Scoop 1/3 cups of batter onto the griddle to make a pancake.
Cook the first side until bubbles cover the pancake surface and begin to pop, then flip the pancake over.
Once the second side has lightly browned (about half the time it takes to cook the first side), remove and serve.