In a large mixing bowl, combine the beaten eggs, milk, flour, and vanilla, and whisk until just blended, being careful to not overmix. Set the batter aside to rest at least 20 minutes.

Position rack in center of oven and preheat the oven to 450ºF.

In a heavy ovenproof 12-inch skillet, melt 2 tablespoons of the butter over high heat. Add the apples, cinnamon, nutmeg, and salt and cook, stirring frequently, until the apples are soft and lightly golden around the edges, about 6 minutes.

Add the brown sugar and cook, stirring, until the apples are caramelized and very soft, 2 to 3 minutes longer. Add the remaining tablespoon of butter and stir to melt.

Working very quickly, pour the batter evenly over the top of the apples. Using oven mitts or pot holders, transfer the skillet to the oven and bake until the pancake is golden brown and puffed, about 15 minutes. Don't be alarmed when you see the edges of the pancake puff up over the top of the pan--this is supposed to happen!

Using oven mitts or pot holders, remove the skillet from the oven and serve the pancake immediately, sprinkled with sifted confectioners' sugar or drizzled with maple syrup, as desired.