Heat the oil in a heavy pot to 350°F, use a thermometer to monitor the temperature during frying as you may need to adjust the heat to keep the temperature constant.
In a small mixing bowl, beat together the egg, buttermilk, vanilla and melted butter.
Sift together the flour, baking powder, and baking soda in a large mixing bowl. Stir in salt and sugar. Add the liquid mixture to the bowl and mix until a thick batter forms.
Use a small cookie scoop to drop dough into the fryer, and fry until golden brown, about 3 - 4 minutes, turning the donuts halfway through cooking. Don’t crowd the pot.
Remove the donuts from the hot oil and rest them on a wire rack placed over a baking sheet to drain excess oil or simply place them on some paper towels to absorb the excess oil.
Serve warm or at room temperature.
To dress up your donut holes, you can roll them in powdered sugar, plain sugar, cinnamon sugar or dip them in a glaze.
For plain sugar and cinnamon sugar topping, roll the donut holes in the sugar while they are still warm.
For powdered sugar and glaze topping, allow the donut holes to cool slightly before rolling, sprinkling or dipping.