Cream together unsalted butter and icing sugar until smooth and fluffy. Incorporate the egg and plain flour, mixing until a dough forms. Shape into a ball, wrap, and chill for 2 hours.

Roll out the chilled dough into a 12-inch circle, approximately ¼ inch thick. Transfer to a tart pan, pressing into edges, and trim excess dough. Chill again while preparing to bake.

Preheat the oven to 350°F (180°C). Prick the base of the tart shell with a fork, line with baking paper, fill with weights, and bake for 20 minutes. Remove weights and bake for an additional 15 minutes until golden.

In a saucepan, combine whole milk, butter, granulated sugar, and vanilla extract. Heat until steaming. In a bowl, whisk together eggs and yolks, then gradually add the hot milk to temper. Return to the saucepan and cook until thickened.

Allow the custard to cool slightly before pouring it into the cooled pastry shell. Cover with an upside-down plate to cool and prevent a skin. Chill in the refrigerator for at least 3 hours.

Top the tart with fresh berries just before serving, evenly distributing them across the custard.