Make the Pastry Cream (Can be done a day ahead):

1. Prepare the milk mixture: In a medium saucepan, combine milk and ¼ cup sugar. If using a vanilla bean, split it lengthwise, scrape out the seeds, and add both seeds and pod to the milk. Heat over medium heat until just simmering, then remove from heat and let steep for 10 minutes. Remove vanilla pod.
2. Whisk the egg mixture: In a medium bowl, whisk together egg yolks, remaining ¼ cup sugar, and cornstarch until smooth and pale yellow, about 2 minutes.
3. Temper the eggs: Slowly pour about ½ cup of the hot milk mixture into the egg mixture while whisking constantly. This gradually raises the temperature of the eggs without scrambling them.
4. Cook the pastry cream: Pour the tempered egg mixture back into the saucepan with the remaining milk. Cook over medium heat, whisking constantly, until the mixture thickens and comes to a boil. Continue cooking for 1-2 minutes while whisking to cook out the starch taste.
5. Finish and chill: Remove from heat and whisk in butter and vanilla extract (if not using vanilla bean). Strain through a fine-mesh sieve into a clean bowl. Press plastic wrap directly onto the surface to prevent a skin from forming. Refrigerate until completely cold, at least 2 hours or overnight.

Make the Choux Pastry:

6. Preheat oven: Position racks in the upper and lower thirds of your oven and preheat to 425°F (220°C). Line two baking sheets with parchment paper.
7. Make the dough: In a medium saucepan, combine water, butter, sugar, and salt. Bring to a rolling boil over medium-high heat, ensuring the butter is completely melted.
8. Add flour: Remove from heat and add flour all at once. Stir vigorously with a wooden spoon until the mixture forms a ball and pulls away from the sides of the pan, about 1 minute.
9. Cook the paste: Return the pan to medium heat and cook, stirring constantly, for 2 minutes. This dries out the dough slightly and cooks the flour. The dough should look slightly shiny and leave a thin film on the bottom of the pan.
10. Transfer and cool briefly: Transfer the hot dough to the bowl of a stand mixer fitted with the paddle attachment (or a large bowl if using a hand mixer). Let cool for 3-4 minutes, stirring occasionally.
11. Add eggs: Beat the dough on medium speed and add eggs one at a time, beating well after each addition. The dough will look separated at first but will come together. After all eggs are incorporated, the dough should be smooth, glossy, and fall from a spoon in a thick ribbon.
12. Pipe the puffs: Transfer dough to a piping bag fitted with a ½-inch round tip. Pipe 1½-inch mounds onto prepared baking sheets, spacing them 2 inches apart (you should get about 24 puffs). Use a damp finger to gently smooth any peaks.
13. Bake: Place both sheets in the oven and immediately reduce temperature to 375°F (190°C). Bake for 20 minutes without opening the oven door. After 20 minutes, rotate the pans and bake for an additional 10-15 minutes until puffs are golden brown and feel light and hollow.
14. Create steam vents: Remove from oven and immediately use a paring knife or skewer to poke a small hole in the side or bottom of each puff to release steam. Return to the turned-off oven with the door slightly ajar for 10 minutes to dry out the interiors. Transfer to a wire rack to cool completely.

Make the Chocolate Sauce:

15. Heat cream: In a small saucepan, bring heavy cream to a simmer over medium heat.
16. Make ganache: Place chopped chocolate in a heatproof bowl. Pour hot cream over chocolate and let stand for 2 minutes. Add butter and corn syrup (if using), then whisk until smooth and glossy. Keep warm until serving.

Assemble the Profiteroles:

17. Finish the filling: When ready to serve, whip the heavy cream to soft peaks and gently fold it into the chilled pastry cream to make crème diplomat.
18. Fill the puffs: Cut each puff in half horizontally or use a piping bag fitted with a small tip to pipe filling through the steam vent hole. If cutting, place a generous dollop of cream in the bottom half and replace the top.
19. Serve: Arrange 3 profiteroles per serving on dessert plates. Drizzle generously with warm chocolate sauce just before serving.