Cut butter into little chunks and keep cold to work into pastry. Whisk flour, sugar, and salt together in a mixing bowl. Add butter and work into the flour until coarsely distributed. Whisk in egg, water, and vanilla; work together with 2 forks until it just barely holds together. Use a piece of plastic wrap to ball up the dough and refrigerate for 20 to 30 minutes.
While the dough is refrigerating, make the filling:
Preheat oven to 375 degrees F.
Melt the butter and sugar together in a 12-inch nonstick oven-proof skillet over medium-high heat. Add the apple slices and stir to coat with melted butter mixture. Sprinkle with cinnamon and cook until apples have softened slightly and take on some color, about 15 minutes. Sprinkle with cinnamon. Smooth the apples out into an even layer in the pan.
Roll out the refrigerated dough on a floured countertop into a 13-inch disk and lay over the top of the apples, trimming any excess. Prick with a fork in several places to let steam release. Brush the egg all over the top of the dough and bake for 40 minutes or until golden browned.
Allow to cool nearly fully before inverting onto a serving platter. Serve with vanilla ice cream or custard.