Preheat oven to 350 degrees F (175 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; 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.
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