Prepare the pastry cream first. Heat the milk until steaming, whisk the sugar, yolks, cornstarch, and salt together, then gradually whisk in the hot milk. Return the mixture to the saucepan and cook over medium heat until thick and bubbling; continue whisking and cooking for about 1 minute.

Remove the pastry cream from the heat and stir in the butter and vanilla. Press plastic wrap directly onto its surface and chill until completely cold.

Make the choux pastry by heating the water, milk, butter, sugar, and salt until the butter melts and the mixture reaches a simmer. Add the flour all at once, stirring vigorously until a smooth dough forms and pulls away from the pan.

Cook the dough for 1–2 minutes to dry it slightly, then transfer it to a bowl and cool for 5–10 minutes. Beat in the lightly beaten eggs gradually until the dough is smooth, glossy, and thick enough to hold a piped shape.

Pipe 4–5-inch logs onto a parchment-lined baking sheet, leaving space between each one. Bake at 425°F (220°C) for 10 minutes, then reduce the oven to 350°F (175°C) without opening the door and bake for another 20–25 minutes.

Turn off the oven, open the door slightly, and leave the éclairs inside for 5–10 minutes. Transfer them to a rack and cool completely before filling.

Make the chocolate glaze by pouring hot cream over the chopped chocolate. Let it rest for 1–2 minutes, then stir until smooth; add the optional corn syrup or butter if you want a shinier, softer finish.

Fill the cooled éclairs through small holes in their bottoms, or split them horizontally and pipe in the pastry cream. Dip the tops in the chocolate glaze and serve once the topping has set slightly.