Whisk together flour, sugar, baking powder, salt, and nutmeg in a large bowl.
In a separate bowl, whisk milk, eggs, melted butter, and vanilla until smooth.
Pour wet ingredients into dry ingredients and mix gently until just combined. Do not overmix.
Cover the dough and refrigerate for 30 minutes to firm up.
Lightly flour a surface and roll dough to ½-inch thickness. Cut into donut shapes.
Heat oil in a deep pot to 350°F (175°C).
Fry 2–3 donuts at a time for 1–2 minutes per side until golden brown.
Remove and drain on a wire rack or paper towels.
Toss warm donuts in sugar or cinnamon sugar if desired.