To make pastry cream: Whisk together sugar, cornstarch, and salt in a medium-size heavy bottomed saucepan. Whisk together milk and egg yolks in a glass measuring cup; add to saucepan along with butter and bring to a boil over medium heat. Let boil 1 minute, still whisking; then remove from heat and stir in vanilla.
Strain pastry cream through a fine-mesh sieve into a bowl. Cover with plastic wrap, pressing it directly onto surface of cream to prevent skin from forming. Refrigerate until chilled, at least 2 hours and up to 2 days. Just before piping into eclair shells, whisk until smooth.
To make the eclair dough: Preheat oven to 425°F. Line 2 baking sheets with parchment or silicone baking mats. Set aside.
In a medium-size saucepan bring water, milk, butter, sugar and salt to a rolling boil.
Remove from heat, mix in flour. Return to burner on medium and stir vigorously with a wooden spoon until the flour begins to pull away from the sides and forms a film on the bottom of the saucepan. The pate a choux should form a ball.
Scrape into a mixing bowl. Using an electric mixer, beat the dough for 1 minute on medium speed to cool slightly before adding 5 eggs one at a time. Stop after each egg to scrape the sides of the bowl. The dough may separate after adding each egg but, will come together when beaten.
Mix until the dough is smooth and glossy and the eggs are completely incorporated.
Use a piping bag fitted with a large plain tip, pipe long fat lengths of dough about 5-6 inches long on baking sheets. Leave at least 2 inches between each eclair. (You should end up with 8-10 eclairs.)
Whisk remaining egg with 1 ½ tsp water. Brush onto tops and sides of eclair dough, using your finger to smooth any tips or bumps.
Bake for 15 minutes, then lower the temperature to 375°F. Continue to bake until puffed up and light golden brown, about 25 minutes more. Try not to open the oven door too often. Let cool on the baking sheet. Use a toothpick or skewer to pierce after baking to allow steam to escape.
Use a medium-size plain pastry tip to make a hole in the end of each eclair. Using a pastry bag fitted with the same size tip, gently pipe the pastry cream into each eclair.
To make the glaze: Heat the cream over medium heat just until boiling. Immediately turn off the heat. Place chocolate in a medium heat proof bowl. Pour hot cream over the chocolate and whisk until melted and smooth. Add corn syrup, mix well. Set aside and keep warm.
Dip tops of eclairs in warm chocolate glaze and set on a sheet pan. You can also frost using an offset spatula or spoon, using the back of the spoon to smooth. Chill, uncovered, at least 1 hour to set the glaze.
Store chilled in an airtight container with wax paper between until serving.