Pour oil into a deep heavy pot until it reaches 2 inches up the sides. Clip a thermometer to the pot and heat over medium-high heat to 350°F. Meanwhile, whisk together flour, sugar, baking powder, salt, and nutmeg in a large bowl.

In a separate bowl, whisk milk, egg, and melted butter until combined. Pour into the dry ingredients and stir with a wooden spoon just until a sticky, shaggy dough forms. Do not overmix - a few lumps are fine.

In a shallow bowl, mix the sugar and cinnamon for the coating. Set aside with the melted butter in a separate small bowl.

Working in batches of 6 to avoid crowding, drop rounded tablespoons of dough into the hot oil. Fry for 2 to 3 minutes, turning occasionally with a slotted spoon, until deep golden brown all over and cooked through. The dough will puff and float when ready. Transfer to a wire rack set over paper towels.

While still warm, roll each donut hole in melted butter, then immediately toss in the cinnamon sugar until evenly coated. Return to the rack and repeat with remaining dough. Serve warm.