In a large bowl, combine the eggs, milk, flour and vanilla extract. Whisk until just blended, being careful not to overmix. Set the batter aside to rest while you prepare the apples (at least 20 minutes).
Preheat the oven to 450°F.
In a heavy, ovenproof (preferably nonstick) 12-inch skillet, melt 2 tablespoons of the butter over high heat. Add the apples, cinnamon, nutmeg, and salt. Cook, stirring frequently, until the apples are soft and lightly golden around the edges, about 6 minutes. Add the 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 potholders, 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 and extend above the top of the inner edge of the pan--this is supposed to happen.
Using oven mitts or potholders, remove the skillet from the oven and serve the pancake, sprinkled with powdered sugar. Drizzle with maple sugar, if desired.