Preheat the oven to 350°F (177°C). Spray a donut pan with non-stick spray. Set aside.
Make the donuts: Whisk the flour, baking powder, baking soda, and nutmeg together in a medium bowl. Set aside.

Whisk the egg, brown sugar, milk, and yogurt together until smooth. Add the melted butter and vanilla, whisking until fully combined. Pour the wet ingredients into the dry ingredients and stir until just combined. Do not overmix. The batter will be very thick.
Spoon the batter into the donut cavities—I highly recommend using a large zipped-top bag for ease, Cut a corner off the bottom of the bag and pipe the batter into each donut cup, filling 2 ∕ 3 –3/4 of the full.

Bake for 9–10 minutes or until the edges are lightly browned. Remove from the pan and bake the remaining donut batter. Allow the donuts to slightly cool before glazing.