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.
Combine the cream, milk, sugar, and vanilla seeds and bean in a medium heavy saucepan and bring to a gentle boil over medium heat. Remove from the heat.
Beat the egg yolks in a medium bowl until frothy and lemon colored, about 2 minutes. Whisk 1 cup of the hot cream into the egg yolks in a slow, steady stream. Gradually add the egg mixture to the hot cream, whisking constantly. Cook over medium-low heat, stirring, occasionally, until the mixture thickens enough to coat the back of a spoon, about 5 minutes.
Remove from the heat and strain through a fine-mesh strainer into a clean bowl. Cover with plastic wrap, pressing it directly against the surface of the mixture to keep a skin from forming. Refrigerate until well chilled, about 2 hours.
Pour mixture into an ice cream maker and freeze according to the manufacturer's instructions. Transfer to an airtight container and freeze until ready to serve.
Yield: 1 quart
Scald the half-and-half and butter in a small heavy saucepan over medium heat. Remove from the heat.
Place the chocolate and vanilla in a medium heatproof bowl. Add the hot half-and-half and let sit for 2 minutes, then whisk until smooth. Serve slightly warm. (The sauce can be kept refrigerated in an airtight container for up to 3 days; rewarm gently before serving.)
Yield: 1 1/2 cups