1. Preheat oven to 375 degrees. Whisk together egg with 1 tablespoon water in a small bowl for the egg wash. Set aside.

2. Line two large baking sheets with parchment paper. Transfer Pate a Choux to a pastry bag fitted with a 5/8-inch plain tip. Pipe dough into 2-inch rounds about 1 inch apart. Gently smooth the pointed peaks with a moistened finger, rounding tops to ensure even rising. Brush tops with reserved egg wash. Bake until puffed and golden brown, about 25 minutes. The profiteroles should feel light and airy. Remove puffs from oven and transfer sheets to wire racks to cool completely.

3. Make the chocolate sauce: Place chocolate in a heatproof bowl over a pan of hot water to melt gently, stirring occasionally with a wooden spoon. Put the milk, cream, and superfine sugar into a pan and bring to a boil, whisking continuously. Take off the heat. Still whisking, pour on the melted chocolate, then return to the heat. Still whisking, let the sauce bubble briefly over medium heat. Turn off the heat and whisk in the pieces of butter, one at a time. Pass the sauce through a fine chinois into a bowl, cover with plastic wrap, and keep warm in a bain-marie.

4. Using a serrated knife, slice off the top third of each puff; set aside. Place a scoop of ice cream on the bottom of each puff; replace tops. Dust with confectioners' sugar and drizzle with chocolate sauce. Serve immediately.