Cut one half cup of butter into small chunks and chill thoroughly.
Combine the flour, powdered sugar, and salt in the food processor. Add the chilled butter and process until the mixture resembles coarse crumbs. Add the egg and pulse for several seconds until the dough comes together. Turn out onto the counter and shape into a disc. Wrap tightly with plastic wrap and refrigerate thirty minutes.
Meanwhile, melt the remaining half cup of butter in 9-inch skillet with the granulated sugar over medium heat, stirring occasionally until the caramel begins to brown and turn amber gold. Increase the heat to medium high and add the apple slices. Saute until slightly transparent, about five minutes. Remove from the heat and arrange the apples in a rosette pattern in the skillet (it may be easiest to transfer most of the slices to a bowl and add them back in as the rosette is formed.) Set aside and preheat the oven to 425°.
Unwrap the chilled dough and sprinkle generously with granulated sugar. Gently roll into a ten to twelve inch circle, using sugar liberally on the work surface to prevent the pastry from sticking. Carefully roll the dough around the rolling pin and transfer to the top of the caramelized apples. Tuck the crust around the edges of the skillet to seal in the apples, folding the excess over.
Bake until the crust is golden brown, about twenty to twenty five minutes. Let the skillet cool for an additional twenty minutes before carefully running a knife around the edge to loosen the pastry. Place the serving plate over the top of the skillet, invert and remove the skillet. Gently rearrange any apples stuck to the skillet back into the pastry. Serve the tarte tatin warm with vanilla ice cream.