Peel, core, and slice the apples into crescent-shaped pieces about a quarter-inch thick.
Melt the butter in a saucepan over medium heat. When it begins to bubble, add the dark brown sugar and cinnamon. Cook while stirring until the sugar is no longer gritty. Add the apples and cook, stirring now and then, until they soften. Add 1/4 cup water and stir to make a syrup around the apples. Reduce the heat to low.
Combine the flour, cinnamon, and white sugar in a small bowl. In a second bowl, combine the egg yolks, vanilla, and milk. Pour the dry ingredients into the wet ingredients and mix with a wire whisk until a few small lumps remain. Add up to 1/4 cup of water to the batter to lighten enough that it pours easily.
Remove the black iron skillet from the oven (with a potholder, of course) and swirl the 1 Tbs. butter around in it until the sides and bottom are coated. Pour the batter into the skillet, about a half-inch from the top of the rim. Return to the oven. Bake for about ten minutes.
Turn the oven up to 425 degrees. Remove the skillet and scoop the apple mixture over the batter. With a fork, force about two-thirds of the the apple pieces down into the batter. Return the skillet to the oven and bake for 20-25 minutes, or until you can actually see the pancake heaving in the pan.
Remove and allow to cool for about two minutes, then loosen the pancake with a knife and slide onto a plate.