In a large mixing bowl, whisk together the flour, sugar, baking powder, nutmeg, and salt.
In a separate bowl, combine the buttermilk, beaten eggs, and vanilla extract.
Gradually add the wet ingredients to the dry ingredients, stirring until just combined. Be careful not to overmix—the dough should be soft and slightly sticky.
Turn the dough out onto a lightly floured surface and gently pat it into a circle about ½-inch thick.
Use a donut cutter (or a round cookie cutter and a smaller cutter for the center hole) to cut out donut shapes. Re-roll the scraps to cut out more donuts.
Place the cut donuts on a parchment-lined baking sheet while you heat the oil.
In a deep skillet or Dutch oven, heat about 2 inches of vegetable oil or shortening to 350°F (175°C). Use a candy thermometer to monitor the temperature.
Carefully add 2-3 donuts to the hot oil, being careful not to overcrowd the pan. Fry for 1-2 minutes per side, or until golden brown.
Use a slotted spoon to transfer the fried donuts to a wire rack or paper towel-lined plate to drain. Repeat with the remaining donuts.
For a simple glaze, whisk together the powdered sugar, milk, and vanilla extract until smooth.
Dip the warm donuts into the glaze, letting the excess drip off, and place them back on the wire rack to set.