Preheat oven to 350°F. Grease two 6-cavity doughnut pans or spray with baking spray.
Stir the coating sugar, cinnamon, and pumpkin spice in a shallow bowl; set aside.
Whisk flour, baking powder, baking soda, salt, pumpkin pie spice, and cinnamon in a large bowl.
Whisk pumpkin, brown sugar, granulated sugar, eggs, buttermilk, vanilla, and melted butter until smooth.
Fold the wet mixture into dry ingredients just until no dry spots remain. Do not overmix.
Transfer batter to a piping bag or zip-top bag. Pipe into pans, filling each well about two-thirds.
Bake 11–13 minutes, until tops spring back and edges look set and lightly golden.
Cool in pans 5 minutes, then turn doughnuts out onto a wire rack.
Brush warm doughnuts with melted butter and toss in the spiced sugar to coat all sides.
Serve warm or let cool completely before storing.