Preheat oven to 425 degrees F. Whisk eggs, milk, sugar, vanilla, salt and cinnamon in a large bowl until well blended. Add flour and whisk until batter is smooth. Place butter in a 13 x 9 inch baking dish in oven until melted, about 5 minutes. Remove dish from oven and place apple slices in overlapping rows on top of butter. Return to oven and bake until apples begin to soften slightly and butter is bubbling, about 10 minutes. Remove from oven and pour batter over apples and sprinkle with brown sugar. Bake for 20 minutes more until puffed and brown.