Place a candy thermometer in a heavy-bottomed pot or Dutch oven, add 2 inches of oil, and heat it to 375°F over medium heat while you make the batter. Line a plate or rack with paper towels.
In a large bowl, lightly beat the eggs, then add the milk and vanilla. Whisk in the flour, granulated sugar, baking powder, and salt until combined.
To pour the batter into the oil, you can use a pastry bag with a round tip, a zipper bag with the corner cut off, a squeeze bottle with a wide opening, or a large funnel. For each cake, squeeze out the batter in overlapping circles (or whatever crazy shape you like). Fry until golden, 45 seconds to 1 minute; flip and cook for another minute or so. Transfer to the prepared plate, dust with confectioners’ sugar, and serve immediately.