Heat about 2 inches of the canola oil in a large skillet over medium-high heat until a deep fry thermometer registers 375 degrees F.
Meanwhile, whisk the flour, baking powder, salt, 1/4 teaspoon cinnamon and 1 tablespoon sugar in a large bowl.
Whisk the eggs, milk, vanilla and mashed bananas in another bowl.
Combine the remaining 1/3 cup sugar and 1 teaspoon cinnamon in a small bowl and set aside.
Whisk the banana mixture into the dry ingredients until just moistened.
Working in batches, carefully drop rounded teaspoons of the dough into the hot oil and fry until golden brown, about 15 to 20 seconds per side.
Remove with a slotted spoon and drain on paper towels.
Roll the beignets in the cinnamon sugar while warm to coat.