Make the Pastry Cream: Fill a large bowl with ice water, then set a fine-mesh sieve over a medium bowl. Set both aside.
In a 3 to 4-quart saucepan, whisk together the sugar, cornstarch and salt. Whisk in the egg yolks until thoroughly combined and all of the dry ingredients have been incorporated. Then, whisk in the milk to combine.
Place the saucepan over medium heat, whisking constantly, until the pastry cream begins to noticeably thicken, about 4 to 8 minutes. When it reaches this point, look for large bubbles to break on the surface. Once they do, whisk constantly for 1 minute, then remove from the heat.
Whisk in the butter and vanilla until melted and completely smooth.
Strain the pastry cream through the fine-mesh sieve into the medium bowl. Immediately press a piece of plastic wrap against the surface of the pastry cream to prevent a skin from forming. Place the bowl of pastry cream into the ice bath to cool for 30 minutes. Transfer to the refrigerator and chill for at least 4 hours or overnight.
Make the Pate a Choux (Choux Pastry): Preheat oven to 350 degrees F. Line two baking sheets with parchment paper.
Place the water, milk, butter, sugar, and salt in a medium saucepan. Place over high heat and bring to a full rolling boil.
Remove from the heat and add the flour all at once. Using a wooden spoon or stiff silicone spatula and stir until the flour is thoroughly mixed in and no lumps remain, pressing and smashing the dough against the sides of the pan.
Return the pot to medium-high heat and cook, stirring frequently, until the dough comes together into one mass and there is a dry film on the bottom and sides of the pan. If you have an instant-read thermometer, it should register 175 degrees F.
Transfer the dough to a mixing bowl and beat on medium speed for about 2 minutes to cool the dough down. It should register 145 degrees on an instant-read thermometer.
Add the eggs one at a time, ensuring each one is fully incorporated before adding the next. Once all of the eggs have been added, scrape down the bottom and sides of the bowl and mix again to ensure everything is fully incorporated, an additional 5 to 10 seconds.
Use immediately or keep at room temperature for up to 2 hours in a piping bag or with a piece of plastic wrap pressed against the surface.
Bake the Eclairs: Transfer the choux pastry to a piping bag fitted with a ½-inch round piping tip (I recommend Ateco #866). Hold the pastry bag at a 45-degree angle and lightly drag the tip along the surface of the parchment paper to create 1-inch wide by 4-inch long lines of pastry dough. Pipe 12 eclairs onto each baking sheet. Dip a finger in cold water and gently pat down any bumps.
Bake the sheets one at a time until the eclairs are puffed, golden brown, and feel light and hollow when picked up, 28 to 32 minutes.
Remove from the oven and while the eclairs are still hot (you may need to use a paper towel to handle them), and using a paring knife, make two ¼-inch holes in the bottom of each eclair (one near each end) by inserting the tip and turning in a circular motion.
Once both sheets have been baked and holes poked in the bottom, return both pans to the still-warm oven. Prop the door open with a wooden spoon and allow the eclairs to set in there for 30 minutes. Remove from the oven and cool completely before filling.
Fill the Eclairs: Place the chilled pastry cream in a pastry bag fitted with a ¼-inch round piping tip (Ateco #802 is perfect for this). Insert the tip of the bag into the holes you created on the bottom of the eclairs, and gently squeeze until filled (the cream will start to come back out of the hole on the bottom). Wipe off excess cream from the bottom and continue until all eclairs are filled.
Alternative Filling Method: You can also slice the eclairs in half horizontally and pipe (I love a large closed star tip for this - Ateco #853) the pastry cream onto the bottom half, and then cover with the top half of the puff. This is a pretty presentation but a bit messier to eat!
Make the Chocolate Glaze: Place the chopped chocolate, butter, corn syrup, and salt in a medium bowl. Microwave in 15-second increments, stirring after each, until completely melted and smooth. Working one at a time, hold each eclair upside down and dip the top in the glaze, then return to the tray. Repeat with all of the puffs. Allow the glaze to set at room temperature, about 20 to 30 minutes. Serve immediately.