Preheat oven to 400°F (200°C). Line a 9x13 inch baking pan with parchment paper, allowing it to extend over the sides for easy lifting.

In a medium saucepan, combine water and ½ cup butter. Bring to a rolling boil over medium heat, stirring until butter is fully melted. Remove from heat.

Add all-purpose flour all at once, stirring vigorously with a wooden spoon until the mixture forms a ball and pulls away from the sides of the pan.

Return to low heat and cook, stirring constantly for 1-2 minutes, to dry out the dough.

Transfer the dough to a large mixing bowl (or the bowl of a stand mixer). Let cool for 5 minutes.

Add eggs one at a time, beating well after each addition until fully incorporated and the dough is smooth, glossy, and thick enough to hold its shape.

Spread the choux pastry evenly into the prepared 9x13 inch pan. Bake for 25-30 minutes, or until golden brown and puffed. Do not open the oven door during the first 20 minutes. Let cool completely on a wire rack.

In a medium saucepan, heat 2.5 cups of milk over medium heat until it just begins to simmer. Do not boil.

In a separate bowl, whisk together the 5 large eggs, granulated sugar, cornstarch, and the remaining 0.5 cup of milk until smooth and no lumps remain.

Gradually temper the hot milk into the egg mixture: slowly pour about ⅓ of the hot milk into the egg mixture, whisking constantly to prevent the eggs from scrambling.

Pour the tempered egg mixture back into the saucepan with the remaining hot milk. Cook over medium-low heat, whisking constantly, until the custard thickens significantly and comes to a gentle boil (it will bubble and pop).

Continue whisking for 1-2 minutes more to ensure the cornstarch is fully cooked.

Remove from heat. Stir in the ¼ cup butter and vanilla extract until fully melted and combined.

Pour the custard into a shallow dish, cover the surface directly with plastic wrap to prevent a skin from forming, and chill in the refrigerator for at least 2-3 hours, or until completely cold and firm.

Once the choux pastry is completely cool, carefully slice it horizontally into two even layers using a long serrated knife.

Place the bottom layer of choux pastry back into the 9x13 inch pan (or onto a serving platter). Spread the chilled vanilla custard evenly over the bottom choux layer.

Carefully place the top choux pastry layer over the custard.

Cover the pan with plastic wrap and chill for at least 1 hour (or up to 4 hours) to allow the custard to set firmly and the flavors to meld.

Before serving, lift the entire dessert out of the pan using the parchment paper handles. Place on a cutting board.

Dust generously with powdered sugar. Slice into 12-16 squares using a sharp knife, wiping the knife clean between cuts for neat presentation.