In a mixing bowl, whisk together 2 cups all-purpose flour, 1/2 cup granulated sugar, 1 tablespoon baking powder, 1/2 teaspoon salt, and 1/2 teaspoon ground nutmeg until evenly combined.

In a separate bowl, beat 2 large eggs, then add 1/2 cup milk, 1/4 cup melted butter, and 1 teaspoon vanilla extract, mixing until smooth.

Gradually add the wet ingredients to the dry ingredients, stirring gently until just combined, being careful not to overmix.

Turn the dough out onto a floured surface and roll it to 1/2 inch thickness.

Use a doughnut cutter to cut out doughnut shapes, rerolling scraps as needed.

Heat vegetable oil in a deep fryer or large pot to 375°F (190°C).

Fry the doughnuts in batches, cooking each side for 1–2 minutes until golden brown, then transfer to a cooling rack.

Allow the doughnuts to cool slightly, then glaze or dust with powdered sugar as desired.