Heat the oven to 200 C / 400 F. Line a sheet pan (33 x 46 cm / 13 x 18 inch) with parchment paper.

Combine the water, cubed butter, salt, and sugar in a 3-quart / 2.8 L saucepan. Bring to a rolling boil over medium heat, stirring until the butter melts completely.

Remove from heat and dump in the flour all at once. Stir hard with a wooden spoon until the mixture pulls away from the sides and forms a ball, about 1 minute.

Return the pan to low heat and cook the dough for 1 to 2 minutes, stirring constantly, to dry it out slightly. Transfer to a mixing bowl and let cool for 5 minutes.

Beat in the eggs one at a time with a hand mixer, making sure each egg is fully absorbed before adding the next. The finished dough should be smooth and glossy, falling off the spoon in a thick ribbon.

Spoon or pipe the dough into 20 mounds, about 4 cm / 1.5 inch wide, spaced 5 cm apart on the sheet pan.

Bake for 20 minutes at 200 C / 400 F, then lower the oven to 180 C / 355 F and bake for another 10 to 15 minutes, until the puffs are deep golden and sound hollow when tapped. Don't open the oven door in the first 20 minutes or the puffs will collapse.

Turn off the oven, prop the door open with a wooden spoon, and let the puffs sit inside for 5 minutes to dry out further. Cool completely on a wire rack.

Whisk the egg yolks, sugar, and cornstarch together in a bowl until pale and smooth.

Heat the milk in a saucepan until it just starts to steam, no need to boil. Slowly pour about a third of the hot milk into the yolk mixture, whisking constantly, then pour everything back into the saucepan.

Cook over medium-low heat, whisking constantly, for 4 to 5 minutes until the custard thickens enough to coat the back of a spoon. Remove from heat and whisk in the butter and vanilla extract.

Pour the custard through a fine-mesh sieve into a clean bowl. Press plastic wrap directly onto the surface and chill for at least 2 hours, until fully cold.

Poke a small hole in the bottom of each cooled puff with a chopstick or piping tip.

Spoon the chilled custard into a piping bag fitted with a 1 cm round tip. Pipe custard into each puff until you feel it fill and weigh down slightly in your hand.

Dust the filled puffs generously with powdered sugar. Serve right away or chill for up to 1 hour before serving.