The pastry puffs can be made ahead and frozen. Freeze them on a baking sheet, and then store them in a zip lock bag. To thaw the puffs, bake them at 350 degrees for five minutes.
For another tasty take, fill the puffs with whipped cream, dust them with confectioners' sugar, and top the dessert with fresh berries.
How to make it
Move the oven rack to a position slightly below the center and heat the oven to 375°. Lightly coat two baking sheets with butter.
In a medium saucepan, bring the water, butter, sugar, and salt to a boil. Remove the pan from the stove top and stir in the flour until well blended. Return the pan to the stove top and cook on medium, stirring constantly, until the mixture pulls away from the sides, about 2 to 3 minutes.
Remove the pan from the stove top once more and transfer the mixture to a medium bowl. Stir in the 4 eggs, one at a time, until each is fully incorporated.
While the dough is still warm, use a tablespoon-size cookie scoop to place 1 1/2-inch-wide mounds on the baking sheets. Stagger the rows to give the puffs room to expand as they bake. Gently brush the top of each with the beaten egg.
Bake the puffs until golden brown, 25 to 28 minutes. (Don't remove them from the oven before they're fully baked, or they'll collapse.) Transfer them to a wire rack to cool completely.
While the puffs are baking, line a baking sheet with waxed or parchment paper. Place 1 1/2-inch-wide ice-cream scoops on the sheet, one for each puff. Cover the ice cream with plastic wrap and freeze until ready to use.
For the chocolate sauce, bring the heavy cream to a boil in a small saucepan. Place the chocolate in a heat-safe bowl and pour the cream on top. Let the mixture sit for 3 minutes, then whisk it until it's smooth. Add the butter, then whisk again.
To assemble the dessert, cut each puff in half, place a scoop of ice cream on the bottom, replace the top, and drizzle with the warm chocolate sauce. Serve immediately.