Place milk and 1/4 cup sugar in a medium saucepan over high heat; cook until almost boiling. In a medium bowl, whisk together egg yolks with remaining 1 tablespoon sugar until thickened. Sprinkle in cornstarch and flour and continue beating until well combined.

While whisking constantly, slowly pour heated-milk mixture into egg-yolk mixture. Pour mixture through a fine sieve back into saucepan, and cook, whisking constantly, over medium-high heat, until it thickens and registers 160°F on an instant-read thermometer, about 2 to 5 minutes.

Transfer to a large bowl and whisk in butter until melted. Add vanilla extract and whisk once more. Cover with plastic wrap, pressing it directly on the surface of the pastry cream; refrigerate until completely cooled. About 1 hour.

In the bowl of a food processor, combine flour and sugar. Add butter, and process until mixture resembles coarse meal, 10 to 20 seconds.

In a small bowl, lightly beat egg yolk; add ice water. With machine running, add the egg mixture in a slow, steady stream through the feed tube. Pulse until dough holds together without being wet or sticky; be careful not to process more than 30 seconds. To test, squeeze a small amount together: If it is crumbly, add more ice water, 1 tablespoon at a time.

Flatten dough and ball into a disk. Wrap in plastic. Transfer to the refrigerator, and chill at least 1 hour.

After an hour, lightly flour a work surface, roll out dough to a 12-inch round, about 1/4-inch thick. Fit dough into a 9-inch round fluted tart pan with a removable bottom, pressing into edges. Using a sharp paring knife, trim dough flush with pan. Chill tart shell until firm, about 30 minutes.

Preheat oven to 350° F. Prick bottom of dough all over with a fork. Line with parchment paper, leaving at least a 1-inch overhang. Fill with pie weights or dried beans. Bake until edges are just beginning to turn golden, about 40 minutes. Remove parchment paper and weights. Whisk together the egg white with a tablespoon of water. Brush evenly on tart shell and return to oven. Continue baking until deep golden all over, 10 to 15 minutes. Cool tart shell completely on wire rack.

Preheat the oven to 350° F. Butter an 9-inch cake pan, line the bottom with a round of parchment paper, and butter and flour the parchment. In the bowl of an electric mixer fitted with the paddle attachment, cream butter and sugar until light in color and fluffy, about 5 minutes. Add vanilla extract, and beat until combined. Add eggs, one at a time, beating until incorporated.

In a medium bowl, stir together flour, baking powder, and salt. Add to egg mixture in three additions, alternating with milk, starting and ending with flour.

Pour the batter into the pan, and bake until the top is golden brown and a tester inserted into the center of the cake comes out clean, 45 to 50 minutes. Cool in the pan for 10 minutes. Turn out onto a wire rack, remove paper, and cool completely. Once cooled slice cake in half.

Place the cooled tart shell onto a plate or platter. Spoon a dollop of the pastry cream and evenly spread it out. Top with a layer of cake. Pour the remainder of the cream and evenly spread it reaching the sides. Arrange fresh berries any way you'd like.

Pour the apricot preserves and water into a small sauce pan and heat until smooth and bubbling. Using a pastry brush, while the glaze is hot, brush on top of the berries. Once the top of the tart is glistening and shiny, you are ready to eat. Slice and serve right away. Place any leftovers in the fridge. Enjoy!