In a large bowl, whisk together the flour, granulated sugar, baking powder, salt, and nutmeg.

Cut in the softened butter until the mixture resembles coarse crumbs.

In a separate bowl, whisk together the eggs, milk, and vanilla extract.

Gradually add the wet ingredients to the dry ingredients, mixing until just combined. Do not overmix.

Cover the dough and refrigerate for at least 1 hour.

On a lightly floured surface, roll out the dough to about 0.5 inch thickness. Cut out doughnuts using a doughnut cutter.

Heat 2-3 inches of oil in a deep pot or Dutch oven to 350°F (175°C).

Carefully fry the doughnuts for 1-2 minutes per side, until golden brown. Fry the doughnut holes as well.

Remove doughnuts from oil and place on a wire rack set over a baking sheet to drain.

In a medium bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth.

Dip the warm doughnuts into the glaze, letting any excess drip off.

Let the glaze set on the doughnuts for about 15-20 minutes before serving.