Make the pastry cream: Whisk the egg yolks, 1/2 cup sugar, 1/4 cup flour, and 1/2 teaspoon kosher salt in a medium bowl until smooth.
Heat 1 1/2 cups milk in a small saucepan over medium heat until small bubbles form at the edges and steam rises, then remove from heat.
Temper the eggs by whisking a little hot milk into the yolk mixture, then whisking in the rest. Return the mixture to the saucepan and cook over medium-low, whisking constantly, until thickened (about 165°F).
Remove from heat, stir in 1 teaspoon vanilla, strain the pastry cream through a fine mesh sieve into a heatproof bowl, press with a ladle to smooth, cover surface with plastic wrap, and chill at least 2 hours.
Preheat the oven to 425°F (220°C) and line baking sheets with parchment paper.
Make the choux: In a saucepan combine 1 cup water, 1/2 cup (1 stick) unsalted butter, 1/2 teaspoon sugar, and 1/4 teaspoon salt. Bring to a simmer over medium-high heat until butter melts.
Add 1 cup all-purpose flour all at once and stir vigorously until the dough forms a ball and a thin film starts to form on the pan bottom, about 5 minutes. Remove from heat.
Transfer dough to a mixing bowl and beat on medium speed. Add the 4 eggs one at a time, fully incorporating each egg before adding the next, until smooth and glossy.
Fit a large piping bag with a jumbo round tip and pipe 4-inch long, 1-inch diameter lines onto prepared parchment, leaving about 3 inches between each.
Bake for 35–40 minutes until puffed, golden brown, hollow, dry, and light. Cool completely on a rack.
Make the chocolate glaze: Heat 1/2 cup heavy cream until just simmering, pour over 4 ounces chopped semi-sweet chocolate, let sit 5 minutes, then whisk until smooth.
Whisk in 1/4 cup light corn syrup and 1 teaspoon vanilla into the glaze until fully combined.
Fill the eclairs: Fit a piping bag with a Bismarck or filling tip, fill it with chilled pastry cream, pierce the bottom of each shell, and pipe in the filling.
Dip the tops of the filled eclairs into the chocolate glaze, allow the glaze to set, then serve.