Whisk the flour, sugar, baking powder, baking soda, salt, and nutmeg together in a large bowl. In a separate bowl, whisk the egg, buttermilk, vanilla, and cooled melted butter until combined.

Pour the wet ingredients into the dry ingredients and stir just until a soft, slightly sticky dough forms. Do not overmix, because excessive stirring can make the finished donuts dense.

Cover the bowl and refrigerate the dough for 30–60 minutes. This short rest firms the dough and makes it much easier to roll and cut.

On a well-floured surface, roll the chilled dough to approximately 1/2 inch (1.25 cm) thick. Cut 3-inch rounds with a donut cutter, or use a round biscuit cutter together with a small cutter for the center.

Heat 1 1/2–2 inches of neutral oil in a heavy saucepan or Dutch oven to 350–360°F (175–182°C). Use a thermometer to monitor the oil, and fry one test donut first to confirm the browning and center doneness.

Carefully lower 2–3 donuts into the hot oil and cook for about 60–90 seconds per side, turning once, until puffed and golden brown. Transfer the cooked donuts to a wire rack with paper towels underneath, then let the oil return to temperature before the next batch.

For the optional glaze, whisk the confectioners’ sugar, milk, and vanilla until smooth. Dip cooled or slightly warm donuts into the glaze and let the excess drip off before serving.