Preheat oven to 180 degrees C. Grease miniature muffin cups or line with paper liners.
Whisk flour, baking powder, nutmeg and salt together in a bowl. Stir milk and vanilla extract together in a separate bowl. Beat butter and 1/3 cup white sugar together in a third bowl using an electric mixer until smooth and creamy then beat in egg.
Stir flour mixture, alternating with milk mixture, into butter mixture until batter is smooth. Spoon batter into a piping bag and pipe into muffin cups, about 2/3-full.
Bake in the preheated oven until a toothpick inserted in the center comes out clean, 10 to 12 minutes.
Mix 1/3 cup white sugar and cinnamon together in a bowl. Add warm doughnuts and gently toss with hands until coated.