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Preheat oven to 350°F.
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Lightly spray a donut pan with non-stick cooking spray and set aside. Whisk together the flour, cinnamon, nutmeg, baking powder and baking soda in a large bowl.
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In a separate bowl, combine the egg, brown sugar, milk and yogurt, then whisk until smooth and sugar is no longer grainy.
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Stir in the melted butter and vanilla, then whisk until fully combined. 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.
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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 ¾ of the way full.
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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.