Preheat oven to 350°F.
Lightly spray a donut pan with non-stick cooking spray and set aside.
In a large mixing bowl, whisk together the flour, cinnamon, nutmeg, baking powder and baking soda.
In a separate bowl for the wet ingredients, combine the egg, brown sugar, milk and yogurt. Then, whisk until smooth and sugar is no longer grainy. Stir in the melted butter and vanilla. Then, whisk until fully combined.
In the large mixing bowl, make a well in the center of the flour mixture and add in the wet ingredients. Mix until just combined, taking care not to overmix.
Place the batter into the donut pan, either spooning the batter in directly or transferring the batter into a pastry bag or Ziploc bag with the corner sliced off to pipe into the donut cups. Fill the pan 3/4 the way full.
Bake for 9-10 minutes until lightly browned. Remove from the oven and cool for 5 minutes on a wire rack before gently removing the donuts from each cavity. Bake any remaining donut batter, repeating the same process.