To prepare the shortcrust pastry: mix the sifted flour with the sugar and salt in a bowl. Add the softened butter (taken out of the fridge at least 20 minutes beforehand) cut into pieces, the yolks and 2 or 3 spoonfuls of water. Mix quickly with a fork, then work with your hands until a smooth ball of dough forms. Cover it with a cotton cloth and put it in the fridge to rest for at least 30 minutes. In the meanwhile prepare the filling: in a bowl, beat the egg whites to stiff peaks using an electric whisk. Add the sugar, the seeds from the vanilla pod (slit it open lengthwise and scrape them out with the flat tip of a knife) and continue beating until the mixture is smooth and homogeneous. Finely chop the almonds in a food processor, tip them into the bowl with the egg whites and mix with the electric whisk. Take the pastry dough out of the fridge and, using your hands, spread it out in the tart tin, which you have greased with plenty of butter and dusted with flour. Make sure it adheres around the side as well (up to a height of about 3 cm). Prick the dough here and there with the prongs of a fork, tip in the filling and level the surface. Wash, peel and core the apples; cut them into thin slices and, starting from the outer edge, arrange them on top of the filling in two concentric circles, overlapping slightly. Sprinkle the sugar over the apple and bake in a preheated oven, at 180 °C, for 35-40 minutes. Let the tart cool down completely before removing it from the tin, then sprinkle it with icing sugar and serve.