In a large bowl, add the egg, mashed banana, milk, and yeast. Mix and let stand for 5 minutes.
Add butter, vanilla, sugar, flour, and salt. Mix until you have a homogeneous dough. On a floured flat surface, knead for 10 minutes, or until the dough is smooth.
Put the dough in a large bowl greased with vegetable oil. Cover with film and let stand for 1-1/2 hours.
Roll the dough on a floured surface half a centimeter thick. Use a 7.5 cm round cutter (can be a glass or a lid) and cut the dough. Place the donuts on a cloth and let stand for 30 minutes.
In a deep saucepan, heat the oil over low heat. Fry donuts for 2 minutes each side, or until golden brown. Set them on paper towels.
Add Nutella in a disposable decorating bag or in a Zip-lock bag and cut the tip. With a knife, pierce each donut. Fill with Nutella. Roll each donut in icing sugar.