Make the tart dough: In a mixing bowl, cream the softened unsalted butter and confectioners' sugar until smooth, 1–2 minutes.
Beat in the egg until incorporated and creamy.
Add the all-purpose flour and fine sea salt; mix on low just until no dry spots remain. Gently knead once or twice to bring the dough together.
Flatten the dough into a disk, wrap, and chill until firm, about 1 hour.
On a lightly floured surface, roll the dough to an 11-inch round, about 1/8 inch thick. Line a 9-inch tart pan with a removable bottom, pressing into corners; trim edges. Dock the base all over with a fork and freeze 15 minutes.
Blind-bake the shell: Preheat the oven to 375°F (190°C). Line the frozen shell with parchment and fill with pie weights. Bake until edges look set, 18–20 minutes. Remove parchment and weights, then bake until the crust is golden and dry, 8–12 minutes more. Cool completely on a rack.
Make the pastry cream: In a medium saucepan, combine the whole milk with the vanilla bean and its seeds. Heat over medium until steaming, then cover, remove from heat, and steep 10 minutes.
In a bowl, whisk the granulated sugar, egg yolks, and cornstarch until pale and smooth, 1–2 minutes.
Return the milk to a bare simmer. Whisking constantly, slowly pour about 1/2 cup hot milk into the yolk mixture to temper. Whisk the tempered mixture back into the saucepan.
Cook over medium heat, whisking constantly, until thick bubbles break the surface and the cream holds lines, 1–2 minutes after it begins to boil. Remove from heat, discard the vanilla pod, and whisk in the 2 tbsp unsalted butter until glossy.
Spread the pastry cream in a shallow dish, press plastic wrap directly onto the surface, and chill until cold and set, 1–2 hours.
Assemble the tart: Stir the chilled pastry cream to loosen and spread it evenly in the cooled crust.
Arrange the strawberries snugly over the cream, cut sides down or fanned as desired.
Glaze: In a small saucepan, warm the apricot jam and water over low heat until fluid; strain if desired. Brush the glaze over the strawberries to coat.
Chill 15 minutes to set the glaze. Slice and serve the same day.