Prepare the dry ingredients: In a mixing bowl, combine 1 cup flour, 1/2 cup sugar, 1/4 tsp baking powder, 1/4 tsp salt, and 1/2 tsp nutmeg. Whisk them together until well blended.
Mix the wet ingredients: In a separate bowl, whisk together 1/4 cup milk, 1 egg, and 1/2 tsp vanilla until smooth.
Combine wet and dry ingredients: Gradually pour the wet mixture into the bowl with the dry ingredients. Gently fold the mixtures together until just combined. Avoid overmixing to keep the doughnuts tender.
Shape the dough: Turn the dough onto a lightly floured surface and roll it out to about 1/2 inch thick. Use a doughnut cutter or two round cutters (one large and one small) to cut out doughnut shapes from the dough.
Heat the oil: In a deep skillet or frying pan, heat enough oil to submerge the doughnuts, at least 2 inches deep. Heat the oil to 350°F (175°C). You can test the temperature by dropping a small piece of dough into it – it should sizzle and rise to the surface.
Fry the doughnuts: Carefully place the cut doughnuts into the hot oil, being cautious to not overcrowd the pan. Fry for about 2-3 minutes on each side, or until golden brown. Use a slotted spoon to flip them, ensuring they cook evenly.
Drain and dust: Once cooked, remove the doughnuts and place them on a plate lined with paper towels to drain excess oil. While still warm, dust generously with powdered sugar.
Serve: Enjoy your warm, freshly made doughnuts plain or with your favorite toppings!