Blend the cold, cubed butter with the flour until the mixture resembles coarse crumbs. Add the powdered sugar, egg, and a pinch of salt, mixing until a smooth dough forms. Shape the dough into a ball, wrap it in plastic wrap, and refrigerate for at least 30 minutes.
Dice the apples and mix them with the lemon juice and breadcrumbs. Heat the mixture in a pan until the apples release some of their moisture. Add the granulated sugar and honey, cooking for a few more minutes, then allow the mixture to cool.
Roll out the pastry dough and line a tart pan with it. Pour the cooled apple filling over the strips for a lattice pattern on top of the tart. Brush the strips with beaten egg. Bake in a preheated oven at 350°F (175°C) for about 30 minutes, or until golden brown.
Heat the milk with the vanilla bean (split lengthwise) until it reaches a boil, then remove from heat and let it infuse for a few minutes. In a bowl, beat the egg yolks with the sugar until the mixture is pale and fluffy. Slowly pour the hot milk into the egg yolk mixture, stirring continuously. Transfer the mixture to a saucepan and cook over low heat, stirring constantly, until it thickens slightly and reaches 180°F (82°C). Do not let it boil.
