In a large saucepan, combine butter, water, milk, and sugar, and bring to a rolling boil.
Add flour and salt all at once, stirring vigorously until the mixture forms a ball and pulls away from the pan.
Remove from heat and let it sit for 5 minutes.
Gradually mix in the eggs one by one, beating well after each addition until the dough is smooth and glossy.
Fit a large star tip into a pastry bag and fill it with the dough.
Chill the dough in the refrigerator for at least an hour.
Heat oil to 375°F (190°C) in an electric skillet or deep fryer.
Cut parchment paper into eighteen 3-inch squares, pipe a 3-inch circle of dough onto each square, and lower them carefully into the hot oil, parchment side down.
Fry for about 1 minute before removing the parchment with tongs.
Continue frying until golden brown, 2-3 minutes per side.
Drain the crullers on paper towels.
For the glaze, whisk together confectioners' sugar, milk, and vanilla until smooth, and dip the tops of the warm doughnuts into the glaze.