Whisk the flour, sugar, baking powder, salt, and nutmeg together in a large bowl. Rub in the cold butter with your fingertips for 2 to 3 minutes, until the mixture looks like coarse sand.
Whisk the egg, milk, and vanilla in a small bowl, then pour it into the flour mixture. Stir until a shaggy dough forms, then knead on a lightly floured counter for 3 to 4 minutes until smooth and soft. If it sticks, add 1 tbsp flour. If it feels dry, add 1 tbsp milk.
Cover the dough with a clean towel and rest it for 15 minutes. While it rests, pour the oil into a 22 cm / 9-inch heavy-bottomed pot so it's about 4 cm / 1.5 inches deep. Heat over medium until a thermometer reads 170 C / 340 F, and line a tray with paper towels.
Divide the dough into 16 pieces of about 55 g each. Roll each piece between your palms into a smooth ball and keep the shaped buns covered with the towel.
Lower 4 buns into the oil with a slotted spoon. Fry for 4 to 5 minutes, turning every minute, until deep golden on every side and a skewer in the center comes out clean. If they brown in under 3 minutes, lower the heat. Lift them onto the tray.
Wait 1 to 2 minutes for the oil to return to 170 C / 340 F, then fry the remaining 3 batches the same way. Cool the buns for 5 minutes before serving warm.