Preheat the oven to 400 degrees F (200 degrees C). Lightly grease the bottom of a deep-dish pie plate.
Beat butter, sugar, and egg in a bowl until creamy. Blend in flour to form a dough, using your hands to knead it together at the end.
Roll pastry out on a lightly floured surface and place into the prepared pie plate so it covers the bottom and the sides.
Beat eggs and sugar in a bowl using an electric mixer. Beat in vanilla extract. Gradually stir in cold milk until fully incorporated. Pour custard mixture over the pastry base and sprinkle with nutmeg.
Bake in the preheated oven until custard is set and a sharp knife inserted near the centre comes out clean, about 45 minutes. Remove from the oven and cool completely before serving, about 1 hour.