In a large bowl, whisk together 2 cups all-purpose flour, 1/2 cup granulated sugar, 2 teaspoons baking powder, 1/2 teaspoon salt, and 1/4 teaspoon nutmeg.

In another bowl, whisk together 2 large eggs, 1/2 cup buttermilk, 1 teaspoon vanilla extract, and 2 tablespoons melted butter until combined.

Pour the wet ingredients into the dry ingredients and stir until a shaggy dough forms, being careful not to overmix.

Cover the dough and refrigerate for at least 30 minutes to 2 hours to chill.

On a lightly floured surface, roll the dough to 1/2-inch thickness, then cut with a 3-inch donut cutter.

Let the cut donuts rest on parchment-lined trays at room temperature for 10 minutes.

Heat 2 quarts of vegetable oil in a heavy pot to 350°F, monitored with a candy thermometer.

Fry 2 to 3 donuts at a time for 90 seconds per side until golden brown, then transfer to a wire rack.

Let the donuts cool for at least 5 minutes before glazing.

In a small saucepan over low heat, whisk 2 cups powdered sugar, 3 tablespoons melted butter, 1/2 teaspoon vanilla extract, and 3 tablespoons milk until smooth, adding more milk if needed, then dip each donut into the glaze and let set.