In a bowl, combine flour, sugar, ground nutmeg or cinnamon, instant yeast, and salt.
In a separate bowl, mix warm milk, melted butter, and egg.
Make a well in the center of the dry ingredients and pour in the wet ingredients. Stir with a wooden spoon, then knead the dough by hand for about 5 minutes or use a stand mixer with a hook attachment.
Place the dough in a greased bowl, cover with plastic wrap and a towel, and let it rise for 1 to 1.5 hours until doubled in size.
Punch the dough to release air bubbles, then roll it out on a floured surface and cut into desired doughnut shapes. Place doughnuts on a baking tray lined with parchment paper, cover with a towel, and let them rise for another 30 minutes.
Heat vegetable oil in a saucepan to 350°F (175°C). Fry 3-4 doughnuts at a time, flipping when golden brown (about 30 seconds per side).
Once fried, you can coat the doughnuts in a sugar glaze by mixing 225g powdered sugar, 75g melted butter, 3-5 tbsp cold milk, and 1 tsp vanilla, or dip them in melted chocolate.