Whisk together the flour, sugar, and salt in a mixing bowl. Cut in the cold butter until the mixture resembles coarse crumbs. Add the egg yolk and 2 tablespoons of ice water, mixing just until the dough comes together. Add the remaining tablespoon of water only if needed.

Shape the dough into a disc, wrap it, and chill for 30 minutes.

Roll the dough on a lightly floured surface and fit it into a 9-inch tart pan. Trim the edges, prick the base with a fork, and chill for 15 minutes while preheating the oven to 375°F.

Line the shell with parchment and fill with pie weights or dried beans. Bake for 15 minutes. Remove the weights and bake for 10 to 12 minutes more, until lightly golden and fully set. Cool completely.

Heat the milk in a saucepan over medium heat until hot but not boiling.

In a bowl, whisk together the sugar, egg yolks, cornstarch, and salt until smooth and pale.

Gradually whisk some of the hot milk into the egg mixture to temper it. Return the mixture to the saucepan.

Cook over medium heat, whisking constantly, until thickened to a smooth pudding-like consistency.

Remove from the heat and stir in the butter and vanilla extract. Transfer to a bowl, press plastic wrap directly onto the surface, and chill until completely cold.

Spoon the chilled custard into the cooled tart shell and smooth the top evenly.

Arrange the strawberries, raspberries, blueberries, and blackberries over the custard in an even layer.

Warm the apricot jam with the water until loosened, then brush lightly over the fruit.

Chill the tart for at least 1 hour before slicing and serving.