In a large bowl, sift together flour, sugar, baking powder, nutmeg, and salt.
Add in milk, egg, butter, and vanilla extract. Stir by hand until ingredients are fully incorporated. It will be sticky. Cover with a damp towel and refrigerate for 1 hour.
Roll out the dough to ¾ inches to 1-inch thick. It is important to make sure the dough is at least ¾ thick or your donuts will be too flat.
Once rolled out, use a donut cutter to cut out shapes. If you do not have a donut cutter, use a round tool like a kitchen glass or a jar lid that is about 3-inches across. You can use a bottle cap or small shot glass for the smaller hole.
Set cut-out donuts on a baking sheet.