In a nonstick wok or medium-large saucepan over medium heat, mix together the water, granulated sugar, salt, butter, and nutmeg. Bring to a boil, then remove the wok from the heat. Using a wooden spoon, stir in the flour, mixing until all of it is incorporated. Set the wok over medium heat and cook, beating continuously, until the dough forms a ball and pulls away from the sides of the pan. Remove from the heat. Transfer the batter to the large bowl of an electric mixer fitted with the dough hook and let cool slightly. Add the eggs, one at a time, beating well on medium speed after each addition. The mixture should be smooth and shiny. Mix in the vanilla.

Clean the wok and dry thoroughly. Pour in the oil and heat over medium-high heat to 375°F. Using a tablespoon, scoop out a rounded spoonful of batter, then use another spoon to release the batter and slide it into the hot oil. Fry 5 or 6 beignets at a time for 3 to 4 minutes, or until golden brown on all sides. Using a slotted spoon, remove the beignets and drain on paper towels.

Sprinkle the beignets generously with confectioners' sugar. Arrange on a serving dish or on individual plates and serve warm.