In a large heavy skillet or deep-fat fryer, heat 3 inches oil to 350 degrees.
Meanwhile, combine flour, sugar and nutmeg, set aside. In medium saucepan heat water or milk, butter and salt over medium heat until boiling and butter is melted. Add vanilla, then add flour mixture all at once. Stir briskly with a wooden spoon until mixture leaves sides of pan and forms a ball. Continue stirring a few minutes more to dry. Remove from heat and beat until slightly cooled. Add eggs, one at a time, beating well after each addition.
Using 2 spoons, shape a generous tablespoon dough into an oblong and drop into hot oil. Fry 3 or 4 at a time, turning beignets as they rise to the surface. Fry 3 to 4 minutes, until golden brown. Drain on paper towels and sprinkle immediately with sifted confectioners' sugar. Repeat with remaining batter.