Preheat the oven to 425°F.
Add the butter to a 9x13-inch baking dish. Place the dish in the oven for a few minutes, just until the butter is completely melted. Keep an eye on it so the butter does not burn.
While the butter melts, prepare the German pancake batter. For the easiest method, add the eggs, milk, flour, and vanilla to a blender. Blend until completely smooth.
If you are mixing the batter by hand, start by whisking the eggs until smooth. Whisk in the flour gradually to create a smooth, thick mixture. Slowly add the milk while continuing to whisk. Once smooth, whisk in the vanilla. Adding the ingredients in this order helps prevent lumps of flour in the batter.
Carefully remove the hot baking dish from the oven. Pour the pancake batter directly over the melted butter.
Return the dish to the oven and bake at 425°F for about 12 minutes, or until the German pancake is puffed and beginning to turn golden around the edges.
Remove from the oven and let cool slightly. The pancake will naturally deflate as it cools.
Cut into pieces and serve with maple syrup, fresh fruit, or your family's favorite pancake toppings.