In a large bowl, mix mashed banana, milk, beaten egg, vegetable oil, and vanilla extract until well combined.
In another bowl, sift together flour, granulated sugar, baking powder, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Heat vegetable oil in a deep pan over medium heat.
Drop spoonfuls of batter into the hot oil, and fry until golden brown, about 2-3 minutes on each side.
Remove the doughnuts from the oil and drain on paper towels.
While still warm, roll the doughnuts in granulated sugar to coat.