Mix dry ingredients: In a bowl, whisk flour, sugar, baking powder, salt, and cinnamon.
Add wet ingredients: Stir in the egg, milk, and melted butter until you get a soft, sticky dough.
Heat oil: Pour oil into a deep pan (about 2 inches deep). Heat to 350°F (175°C).
Shape and fry: Scoop small spoonfuls of dough and carefully drop them into the hot oil. Fry in batches, turning until golden-brown (about 2–3 minutes each side).
Drain and coat: Remove with a slotted spoon. Drain on paper towels. Roll warm donut holes in sugar or cinnamon sugar.