In a large bowl, whisk together flour, sugar, baking powder, salt, and nutmeg.
In another bowl, combine milk, eggs, and vanilla extract.
Pour the wet ingredients into the dry ingredients and mix until just combined.
Stir in the melted butter until smooth.
Heat oil in a deep fryer or large pot to 375°F (190°C).
Using a doughnut cutter, cut out shapes from the dough.
Carefully drop the doughnuts into the hot oil, a few at a time.
Fry until golden brown, about 1-2 minutes per side.
Remove doughnuts and place them on paper towels to drain excess oil.
Once cool, glaze or dust with powdered sugar as desired.