Whisk together the flour, sugar, and salt in a large bowl. Cut in the cold butter until the mixture resembles coarse crumbs.

Add the egg yolk and 2 tablespoons of ice water. Mix gently until the dough starts to come together, adding the remaining tablespoon of water only if needed.

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

Preheat the oven to 375°F. Roll out the chilled dough and fit it into a 9-inch tart pan. Trim the edges, prick the base with a fork, and chill for 10 minutes.

Line the shell with parchment and fill with pie weights. Bake for 15 minutes. Remove the weights and parchment, then bake for 10 to 12 minutes more until golden. 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.

Slowly pour the hot milk into the egg mixture while whisking constantly, then return the mixture to the saucepan.

Cook over medium-low heat, whisking constantly, until the custard thickens to a smooth pudding-like consistency.

Remove from the heat and stir in the butter and vanilla. Transfer to a bowl and press plastic wrap or parchment directly onto the surface. Cool until no longer warm.

Spread the cooled custard evenly into the cooled tart shell.

Arrange the sliced strawberries, raspberries, blueberries, and blackberries evenly over the custard.

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.