Heat the milk in a small saucepan over medium heat, just until bubbles start to appear across the surface. Add the butter to the pot and remove it from the heat. Let it cool to 105 degrees.
Add the following ingredients to your bread machine, in the order suggested by your manufacturer: the cooled milk and butter, flour, yeast, salt, sugar, egg, bananas, and vanilla. Process on the dough cycle.
On a floured board, roll the finished dough out into a quarter inch thick rectangle. Cut the dough into 32 two inch squares. Place the squares on 2 parchment lined, flour dusted baking sheets. Cover the pan loosely with plastic wrap and let it rise in a warm place for 30 minutes.
In an enameled cast iron Dutch oven, or other large pot, heat two inches of vegetable oil to 350 degrees. Fry the beignets, about four at a time, on both sides, for about three minutes, or until they're golden brown. Be sure and maintain a constant 350 degree temperature. Drain over a wire rack.
If you prefer, bake the beignets at 400 degrees for 12 minutes.
Dust the filled beignets with the powdered sugar and serve right away.