In large mixing bowl combine the beaten eggs, milk, flour, and vanilla, and whisk until just blended, being carful not to overmix. Set the batter aside to rest at least 20 minutes.
preheat oven to 450
In heavy oven proof 12 inch skillet, melt 2 TBSP 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 TBSP of butter and stir to melt.
Working very quickly, pour the batter evenly over the top of the apples. Place the skillet in the over and bake 15 minutes.