In a large bowl, combine flour, sugar, baking powder, salt, and nutmeg.
In another bowl, whisk together milk, eggs, melted butter, and vanilla extract.
Pour the wet ingredients into the dry ingredients and mix until just combined.
Heat oil in a deep fryer or large pot to 375°F (190°C).
Roll out the dough on a floured surface to about 1/2 inch thick.
Use a donut cutter to cut out donuts and holes.
Carefully place donuts in the hot oil, frying them in batches.
Fry for about 1-2 minutes on each side until golden brown.
Remove from oil and drain on paper towels.
Allow to cool slightly before glazing or topping as desired.