In a food processor or large bowl, combine the flour, sugar, and salt and pulse or whisk to blend.
Add the cold cubed butter and pulse or cut in until the mixture looks like coarse crumbs with pea-sized pieces of butter remaining.
Add the egg yolk and 2 tablespoons of ice water and pulse just until the dough starts to come together. Add remaining tablespoon if needed.
Turn the dough out onto a lightly floured surface, gather it into a disk, wrap in plastic wrap, and refrigerate for at least 1 hour.
Preheat oven to 375°F. Roll the chilled dough into a 12-inch circle about 1/8 inch thick on a floured surface.
Transfer the dough into a 9-inch tart pan, pressing it into the bottom and fluted sides. Trim excess and prick the bottom with a fork.
Line the crust with parchment paper, fill with pie weights, and bake for 15 minutes. Remove weights and bake 10-12 minutes more until golden.
Let the crust cool completely in the pan on a wire rack before filling.
In a medium saucepan, heat the milk and half the sugar (1/4 cup) over medium heat until steaming but not boiling.
In a separate bowl, whisk together the egg yolks, remaining 1/4 cup sugar, cornstarch, and salt until smooth and pale.
Slowly pour about a third of the hot milk into the egg mixture while whisking constantly to temper the eggs.
Pour the tempered mixture back into the saucepan with the remaining milk, whisking constantly.
Cook over medium heat, whisking constantly, for 3-5 minutes until the mixture thickens to a pudding-like consistency.
Remove from heat and stir in the vanilla bean paste and butter until smooth and glossy.
Pour the pastry cream through a fine mesh sieve into a bowl, press plastic wrap directly onto the surface, and refrigerate for at least 1 hour.
Once the crust and pastry cream are fully chilled, whisk the pastry cream briefly to loosen it.
Spread the pastry cream evenly into the cooled crust using an offset spatula, smoothing the top.
Arrange the peach slices in a circle around the outer edge, slightly overlapping, and fill the center with halved strawberries cut side down.
Warm the apricot preserves with the water until melted and smooth. Strain through a sieve to remove fruit chunks.
Using a pastry brush, gently brush the glaze over the strawberries and peaches to give them a glossy finish.
Refrigerate the assembled tart for at least 30 minutes to set before slicing. Garnish with fresh mint leaves if desired.