Heat the oven to 350°F (177°C). Lightly coat a donut pan with nonstick cooking spray. Because the batter makes 8 donuts, grease two wells in a second pan if you have one, or bake the extra batter in a second round. Set the pan aside.

In a large bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, and nutmeg. Set the dry mixture aside.

In a medium bowl, whisk the egg, brown sugar, milk, and sour cream or yogurt until smooth. Add the melted butter and vanilla and whisk until fully combined. Pour the wet ingredients into the dry ingredients and stir just until no streaks of flour remain; do not overmix. The batter will be thick.

Scrape the batter into a large zip-top plastic bag. Cut off a corner with scissors and pipe the batter into the prepared donut molds, filling each mold about half to two-thirds full.

Bake for 9 to 10 minutes, until the donuts spring back when gently pressed. Cool the donuts in the pan on a wire rack for 2 minutes, then remove them from the pan and place them on the rack to cool slightly. If you have extra batter, re-grease the pan and bake the remaining batch.

In a shallow bowl, combine the granulated sugar and cinnamon. Dip each slightly cooled donut into the melted coating butter, then roll it in the cinnamon sugar until evenly coated.

Serve the donuts as soon as possible after coating. Store any leftovers in an airtight container at room temperature for 1 to 2 days, or in the refrigerator for up to 1 week.