In a large Dutch oven or heavy-bottomed pot, heat the vegetable oil to 350°F (175°C), using a thermometer to monitor the temperature.

While the oil heats, whisk together the flour, granulated sugar, baking powder, salt, and nutmeg in a medium mixing bowl until well combined.

In a separate small bowl, whisk together the milk, egg, melted butter, and vanilla extract until smooth.

Pour the wet ingredients into the dry ingredients and stir gently with a spatula just until the batter comes together; be careful not to overmix or the donuts will be tough.

Once the oil is at temperature, carefully drop rounded tablespoons of batter into the hot oil, frying about 4 to 5 at a time to avoid overcrowding the pot.

Fry the donut holes for about 2 to 3 minutes, flipping them halfway through, until they are puffed and deep golden brown on all sides.

Remove the fried Timbits with a slotted spoon and place them on a paper towel-lined plate to drain excess oil.

While the donuts cool slightly, prepare the glaze by whisking the powdered sugar and 3 tablespoons of milk in a small bowl until smooth and runny.

Dip each warm donut hole into the glaze, turning to coat evenly, then place on a wire rack to let the excess glaze drip off.

Serve the glazed Timbits immediately while they are still warm and the coating is slightly sticky.