For the pastry: Preheat oven to 425º. Grease a baking sheet with 1⁄2 tbsp. of the butter, dust with 1 tbsp. of the flour, tapping out excess, and set aside. Put milk, 1⁄4 cup water, granulated sugar, salt, and the remaining butter into a medium heavy-bottomed saucepan and bring to a boil over high heat. Remove pan from heat and beat in all of the remaining flour at once with a wooden spoon. Then return pan to medium heat and vigorously beat with the spoon until mixture forms a thick dough and pulls away from the sides of the pan, about 2 minutes. Transfer dough to a medium mixing bowl. Beat in one-third of the eggs at a time, beating constantly with the spoon after each addition, until dough is smooth.
Transfer dough to a pastry bag fitted with a plain round 1⁄2" tip. Pipe 12 mounds of dough about 1 1⁄2" wide onto the prepared baking sheet about 1" apart (you may have enough dough to make 1 extra mound; cook's treat), then dust mounds of dough with confectioners' sugar. Transfer to oven, immediately reduce oven temperature to 400º, and bake until puffed and deep golden brown, 15–18 minutes. Set puffs on a rack to cool completely.
For the sauce: Melt chocolate in a medium bowl set over a medium pot of gently simmering water over medium-low heat. Meanwhile, bring milk to a simmer in a medium saucepan over medium heat. Add cream to milk and return to a simmer. Add the melted chocolate, sugar, and butter and whisk until smooth. Keep sauce warm over lowest heat.
To assemble the puffs, make a wide incision in the side of each puff. Fill each puff with a small scoop of ice cream. Spoon warm sauce over the puffs. Serve immediately.