Prepare your baking sheets by lining them with parchment paper and preheating your oven to 400°F (200°C). Draw 12 4-inch lines on the parchment as piping guides, then flip the paper over so the ink doesn’t touch the pastry.
Make the choux pastry by combining water, butter, sugar, and salt in a medium saucepan over medium heat until the butter completely melts and the mixture comes to a rolling boil.
Add the flour all at once and stir vigorously with a wooden spoon until the mixture forms a ball and pulls away from the sides of the pan. Continue cooking and stirring for another 1-2 minutes to dry out the dough.
Transfer the dough to a bowl and let it cool for about 5 minutes until it’s warm but not hot to the touch – this prevents cooking the eggs when added.
Incorporate the eggs one at a time, beating thoroughly after each addition until the dough is smooth and glossy. The correct consistency is reached when the dough falls from the spoon in a V-shape.
Fill a pastry bag fitted with a large round tip with the choux dough and pipe 4-inch long strips onto the prepared baking sheets, leaving 2 inches between each eclair.
Bake for 25-30 minutes until the eclairs are puffed, golden brown, and feel hollow when tapped. Do not open the oven door during the first 20 minutes of baking.
Remove from oven and immediately poke a small hole in one end of each eclair to release steam. Allow to cool completely on a wire rack.
Prepare the pastry cream by heating milk and half the sugar in a saucepan until simmering. Meanwhile, whisk egg yolks with remaining sugar and cornstarch in a bowl until pale.
Temper the egg mixture by slowly pouring about 1/3 of the hot milk while whisking constantly, then pour this mixture back into the remaining hot milk in the saucepan.
Cook the pastry cream over medium heat, whisking constantly, until it thickens and comes to a boil. Continue cooking for 1-2 minutes more to cook out the cornstarch flavor.
Remove from heat and stir in the vanilla and butter. Transfer to a bowl, press plastic wrap directly onto the surface to prevent a skin from forming, and refrigerate until cold.
Make the chocolate glaze by heating the cream until just simmering, then pour over the chopped chocolate. Let sit for 2 minutes, then stir until smooth. Add butter and corn syrup, stirring until glossy.
Assemble the eclairs by cutting three small holes in the bottom of each shell with a small knife. Fill a pastry bag fitted with a small round tip with the pastry cream and pipe into each hole until the eclair feels heavy.
Dip the top of each filled eclair into the chocolate glaze, allowing excess to drip off. Place on a wire rack to set for at least 30 minutes before serving.