Preheat oven to 400 degrees F.
To make the pastry, let the butter soften to room temperature and put it into a mixer on low speed. Pulse for 2 seconds before adding the egg followed by the water. Mix for a few seconds and then add the salt and the flour, keeping 2 tablespoons aside to add later in case the dough is too sticky. Work fast because the gluten in the flour makes the dough go elastic.
Stop the mixer before the dough turns into a ball. Flatten the pastry and shape a circle about 6 inches wide. Place the pastry on a plate, wrap it in plastic wrap and leave in the refrigerator for 1 hour minimum to 24 hours. This lets the gluten relax and when you roll out the pastry it stays flat.
Caramel Apple Filling: Cut the butter into little bits and scatter on a 10-inch baking dish. Shake the sugar over it and add 2 tablespoons of water to keep it from crystallizing. Caramelize the sugar by placing the dish on a medium heat. Meanwhile, peel the apples, cut in 2 and remove the core. Cut them into 4 big pieces. Once the butter and sugar have caramelized take the pan off the heat. Place the pieces of apple vertically on top of the caramel in the baking tin, taking care to fill the gaps with more pieces so they stick together in a solid mass. Sprinkle 2 tablespoons of sugar over the apples. Put the dish back on a heat diffuser for 15 minutes at a medium heat until the caramel starts bubbling up through the apples.
Preheat the oven to 400 degrees F. Roll the pastry out and place over the apples folding it in at the edges. Make 3 or 4 holes with a knife and 1 in the middle to let steam out when baking. Bake for 20 minutes and then let it rest for 15 minutes. It is important that you do this otherwise the apples will fall apart when you turn it over. Take a dinner plate and put it over the baking tin. Turn it over. Slowly remove the baking tin.
Serve lukewarm with vanilla ice cream, heavy cream or whipped cream.