Whisk the milk and half the sugar in a 2-quart / 2 L saucepan and heat over medium until steaming, about 4 minutes.

Meanwhile, whisk the egg yolks, remaining sugar, and cornstarch in a bowl until pale and smooth.

Slowly pour a third of the hot milk into the yolk mixture, whisking constantly, then pour it all back into the saucepan.

Cook over medium-low heat, whisking constantly, until the cream thickens and holds a ribbon trail off the whisk, about 3 to 4 minutes at a gentle boil.

Remove from heat and stir in the vanilla and rum. Press plastic wrap directly onto the surface and chill for at least 2 hours, until fully cold.

Beat the butter and sugar in a stand mixer until light and fluffy, about 2 minutes.

Add the egg yolks one at a time, then mix in the rum, vanilla, and lemon zest.

Whisk the flour, baking powder, and salt together, then add to the butter mixture in two additions, mixing just until a soft dough forms.

Divide the dough in half, flatten into disks, wrap, and chill for 2 hours, until firm enough to roll.

Heat the oven to 180 C / 355 F. Butter a 23 cm / 9-inch springform pan.

Roll one dough disk between two sheets of parchment to fit the base and sides of the pan, about 5 mm thick. Press into the pan, patching any cracks with your fingers.

Spread the chilled pastry cream evenly over the dough, leaving a 1 cm border.

Roll the second disk and lay it over the filling, pinching the edges to seal.

Brush the top with egg wash, then score a crosshatch pattern with a fork or the tip of a knife.

Bake for 40 to 45 minutes, until deep golden brown all over.

Cool in the pan for 30 minutes, then release and cool completely on a rack for at least 2 hours before slicing.