Make the dough: In a bowl whisk together the All-Purpose Flour, Baking Powder and Fine Salt.
In a separate large bowl beat room-temperature Unsalted Butter with Granulated Sugar until pale and fluffy, then beat in the Large Eggs one at a time and the finely grated zest of the Lemon.
Fold the dry ingredients into the butter mixture until a soft dough forms, taking care not to overwork it; shape into a disk, wrap in plastic and chill for at least 1 hour.
Make the pastry cream: Split the Vanilla Bean and scrape the seeds, then warm the Whole Milk with the seeds and pod (and a little lemon peel if you like) until it just begins to simmer; remove from heat and let infuse for 10 minutes.
Whisk the Egg Yolks with a bit more Granulated Sugar and the Cornstarch until smooth and pale. Temper the egg mixture with a few spoonfuls of the hot milk, then return everything to the pan and cook over medium heat, whisking constantly, until the mixture thickens to a pudding-like consistency.
Remove from heat, stir in a knob of Unsalted Butter and the Dark Rum for flavor, then transfer the pastry cream to a bowl, press plastic directly on its surface to prevent a skin, and chill until cool.
Assemble the cake: Divide the chilled dough into two pieces (about 60/40). Roll the larger piece between parchment into a circle and use it to line a buttered 9-inch (23 cm) tart pan with a removable bottom, trimming excess.
Spread the cooled pastry cream evenly over the dough-lined pan, leaving a small border. Roll the remaining dough and place it over the filling, seal and flute the edges, then decorate the top with a fork or a shallow crisscross pattern.
Make an egg wash by beating the remaining Egg Yolks with a splash of Water and brush the surface lightly; chill the assembled cake for 15–30 minutes in the fridge to firm up the top.
Bake in a preheated oven at 350°F (175°C) for 35–45 minutes, until the crust is deep golden and the filling is set. Let cool completely in the pan before unmolding—Gâteau Basque slices best at room temperature.
