Preheat the oven to 425 degrees F. Line a large baking sheet with parchment paper and set aside.

Combine the milk, butter, sugar, and salt in a medium saucepan and bring to a slow boil over medium heat and cook until the butter is melted. Remove from the heat and add the flour all at once, stirring with a heavy spoon. Return the pan to medium-low heat and stir briskly until the mixture becomes a smooth paste and pulls away from the sides of the pan in a ball, 1 to 2 minutes.

Remove from the heat and let cool slightly, 1 to 2 minutes. Stirring constantly with a heavy wooden spoon, add 4 of the eggs, 1 at a time, beating well after the addition of each, and mix until smooth. Let the dough cool.

Beat the remaining egg with the water in a small bowl to make an egg wash.

Transfer the dough to a pastry bag fitted with a plain 3/4-inch tip and pipe out 16 ping-pong-ball-sized rounds onto the prepared sheet, about 2 inches apart. (Alternatively, spoon onto the baking sheet with a large spoon.) Lightly brush the egg wash on the tops and sides of the dough using a pastry brush and bake until doubled in size, 14 to 15 minutes. Reduce the temperature to 375 degrees F and bake until the balls are golden brown, firm to the touch, and hollow inside, 6 to 7 minutes. Turn off the oven and leave the oven door ajar for 10 minutes for the puffs to dry.

Cool completely on a wire rack. Cut each profiterole in half and scoop out the doughy centers with a small spoon.

To serve, fill the bottom halves with a small scoop (about 1-ounce) of the ice cream and cover with the tops. Place 2 to 4 profiteroles on each dessert plate and sift the confectioners' sugar over the profiteroles and decoratively onto the plates. Drizzle each profiterole with warm chocolate sauce and serve immediately.