1. Brush the lemon under hot water, put the sugar, vanilla, and butter in a bowl, grate the lemon zest over it and mix with a spatula until the mixture whitens.
2. Add the whole egg and one yolk (the other will be for glazing) and a pinch of salt, mix and gradually add the flour.
3. Work with your fingertips until the dough is smooth and perfectly homogeneous.
4. Roll it into a ball and let it rest for 2 hours in the refrigerator.
5. Meanwhile, prepare the pastry cream: pour the milk into a saucepan and bring to a boil.
6. Put the yolks and sugar in a bowl.
7. Work them until the mixture whitens.
8. Add the flour, mix, and pour in the rum.
9. Gradually pour in the boiling milk.
10. Pour this mixture into a saucepan, place it over low heat and cook for about 10 minutes, stirring constantly.
11. When the cream starts to boil, wait another 1 minute before removing the saucepan from the heat.
12. Pour the cream into a bowl and let it cool completely.
13. Stir it occasionally to prevent a crust from forming on the surface.
14. Remove the dough from the refrigerator 10 minutes before preparing the cake.
15. Preheat the oven to 350°F (180°C).
16. Butter a 10-inch (26 cm) tart pan.
17. Take 2/3 of the dough, roll it out, and line the pan with it (it should extend 0.6 inches (1.5 cm) beyond the edge of the pan).
18. Fill with pastry cream.
19. Roll out the remaining dough (which should also extend beyond the tart pan) and place it on top.
20. Seal the edges by rolling them.
21. Beat the remaining egg yolk with a tablespoon of water, brush it over the entire surface of the cake, draw lines with the tip of a knife, and prick the top in places to prevent air pockets from forming during baking.
22. Bake for 35 minutes.
23. The cake should be golden brown.
24. Let it cool before unmolding.
25. Serve cold.