Preheat the oven to 350 degrees F (175 degrees C). Grease donut baking pans or miniature muffin cups.

To make the donuts: Whisk flour, baking powder, nutmeg, and salt together in a bowl. Stir milk and vanilla extract together in a separate bowl. Beat butter and 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 donut pans or muffin cups.

Bake in the preheated oven until a toothpick inserted in the center comes out clean, 10 to 12 minutes.

To make the coating: Mix sugar and cinnamon together in a bowl.

Add warm donuts to cinnamon-sugar mixture; gently toss with hands until coated.