In a large mixing bowl, whisk together the flour, 1/2 cup sugar, baking powder, salt, and nutmeg until well combined.
In a separate bowl, whisk together the milk, eggs, vanilla extract, and melted butter until smooth.
Pour the wet ingredients into the dry ingredients and mix until just combined. Do not overmix; a few lumps are okay.
Heat about 2 inches of vegetable oil in a deep pot or fryer to 350°F (175°C).
Using a spoon or a small ice cream scoop, drop spoonfuls of the dough into the hot oil, being careful not to overcrowd the pot. Fry until golden brown, about 2-3 minutes per side.
Remove the doughnuts with a slotted spoon and drain on paper towels.
In a shallow bowl, combine the 1/2 cup sugar and cinnamon (if using). While the doughnuts are still warm, roll them in the sugar mixture until fully coated.
Allow the doughnuts to cool slightly before serving.