Preheat the oven to 425.
Make the pancake batter: In a large bowl whisk together eggs, melted butter, milk and vanilla. Add flour, sugar, baking powder and salt and mix well to combine. Try to whisk out most of the lumps, but it's ok if some remain.
Melt 2 tbsp butter in an 8 or 10 inch ovenproof skillet (such as cast iron) over medium low heat. Swirl to cover the bottom and sides of the pan. Add the white and brown sugars, cinnamon, allspice and nutmeg. Stir. The mixture will look dry at first but will loosen as the sugars melt.
Add the sliced apples to the pan and cook, stirring with the sugar, for two minutes.
Spread the apples out into an even layer and add the batter mixture. Turn the heat off the pan and move the pan to the oven. Bake for 15 minutes.
Lower the oven temperature to 350 and bake for an additional 10 minutes.
Remove from the oven and let sit for 5 minutes. Flip the pancake by running a knife around its edges to loosen it from the pan and placing a plate or cutting board larger than the pan on top of it. Use an oven mitt and flip in one smooth motion. The pancake should release from the pan easily.
Top with powdered sugar or maple syrup and serve.