Pour the milk into a medium saucepan over medium heat and warm until it barely begins to steam, avoiding boiling. In a medium bowl, whisk together egg yolks and sugar until light and well combined. Add cornstarch and whisk vigorously to dissolve lumps. Gradually whisk a ladle of the hot milk into the egg mixture to temper, then whisk in remaining milk. Return the combined mixture to the saucepan and cook over medium-high heat, whisking constantly until thickened and just starting to boil.

Remove from heat and stir in cold butter and vanilla extract. Strain the custard through a sieve into a clean bowl to ensure smoothness. Cover surface directly with plastic wrap to prevent skinning and chill for at least 2 hours or preferably overnight.

Preheat oven to 425°F (218°C) and line a sheet pan with parchment paper. In a large saucepan over medium-high heat, combine water, butter, salt, and sugar. Bring to a rolling boil then immediately remove from heat. Quickly add all the flour at once and whisk vigorously for about 30 seconds until combined.

Return pan to medium heat and cook, stirring constantly with a silicone spatula for 30 seconds until dough is dryer and sticks to the bottom slightly.

Transfer dough to a stand mixer with paddle attachment or use a hand mixer. Beat on medium speed for 30 seconds. Add 3 eggs one at a time, mixing well after each until dough is smooth, glossy, and thick but falls in a V-shaped ribbon from paddle. If too stiff, lightly beat remaining egg and add gradually to achieve correct consistency.

Fit a pastry bag with a large plain tip and pipe 10-12 logs (about 1-inch wide by 4-inches long) onto the parchment-lined baking sheet, spacing at least 2 inches apart. Whisk remaining egg with 2 teaspoons water and brush over the dough. Smooth any uneven areas gently with your fingers.

Bake at 425°F (218°C) for 15 minutes, then reduce heat to 375°F (190°C), rotate the pan, and bake for an additional 25 minutes until puffed and golden brown. Cool completely on the baking sheet.

Freeze cooled shells until hard, then store in an airtight container for up to 1 month. Thaw to room temperature and refresh in a 350°F oven for 3-5 minutes before glazing and filling.

Place chopped chocolate in a heatproof bowl. Microwave heavy cream until just boiling (about 30 seconds). Pour hot cream over chocolate and stir until melted and smooth. Stir in vanilla extract and corn syrup if using.

Slice each éclair lengthwise in half. Dip tops into the chocolate glaze and let excess drip off. Place glazed tops on parchment-lined sheet. Fill bottom halves with chilled pastry cream using a spoon or pastry bag. Once glaze sets, sandwich tops over fillings. Serve immediately or refrigerate covered for up to 2 days.