Preheat oven to 375°F (190°C). Lightly grease a 6-cavity doughnut pan and set aside.
In a small bowl, stir together flour, baking powder and nutmeg.
In a medium bowl, combine sugar and ¼ cup melted butter. Stir in the flour mixture until just combined.
Spoon the batter into the prepared doughnut pan, filling each cup half-full. (Be careful not to cover the center hole.) Bake in the preheated oven for 12 to 15 minutes, or until tops are golden brown.
Remove doughnuts from oven and let cool for 5 minutes before removing the doughnuts from their cups. Let cool 5 more minutes on a wire rack.
Stir together the cinnamon and sugar in a shallow bowl until combined. Melt ¼ cup butter. Working one at a time, quickly dip each doughnut into the melted butter, then roll the doughnut in the bowl of cinnamon sugar and coat thoroughly. Gently shake off any excess sugar. Set on a wire rack until ready to serve.