Whisk the eggs in a large bowl. Add the banana, flour, milk, sugar, baking powder, and salt, and whisk well to make a smooth batter.
Pour enough oil to come halfway up the sides into a large deep pot or electric deep fryer and heat over high heat to 360°F.
In batches, without crowding, drop heaping tablespoons of the batter into the hot oil (you can do 6 or 7 beignets at a time) and fry, turning them to brown evenly, about 2 minutes. Using a slotted spoon, transfer to paper towels to drain.
Sift the confectioners’ sugar through a sieve onto the beignets and serve warm.
Yield: 18 to 24 beignets