Prepare an unbaked tart crust in a 9-inch (23 cm) dish using the Tart Crust By Hand recipe linked in the notes. Chill the lined dish while making the filling.
Lightly beat the eggs. Mix the sugar and cornstarch separately, then whisk in the beaten eggs.
Heat the milk over low heat. Split the vanilla bean lengthwise, scrape out the seeds and add the seeds and pod to the milk. Bring to a simmer.
Gradually add half the hot milk to the egg mixture, stirring constantly. Remove the vanilla pod from the pan, then return the egg mixture to the remaining milk.
Whisk over low heat until thick enough to coat the back of a spoon. Pour into the prepared tart crust.
Bake in a preheated oven at 400°F (200°C) for 30–40 minutes, then broil for 2–3 minutes, watching closely, to brown the top. Let cool and set before removing from the dish and slicing.