Heat the oven to 350°F (175°C). Grease both donut pans carefully, including the center posts, and place them on a level surface.
Whisk flour, baking powder, salt, cinnamon, and batter sugar in a bowl. In another bowl whisk eggs, milk, cooled butter, and vanilla.
Fold wet ingredients into dry just until combined. Avoid beating the batter smooth after all the flour has been moistened.
Transfer to a piping bag or zip-top bag with one corner snipped. Pipe into the cavities about two-thirds full, keeping center posts exposed.
Bake 10–14 minutes until the donuts spring back and a tester shows no wet batter. If baking pans on separate racks, switch positions once near the end.
Cool in the pans for 5 minutes, then loosen carefully and turn onto a rack. Meanwhile mix coating sugar and cinnamon in a shallow bowl.
Brush warm donuts lightly with melted coating butter. Dip or roll in cinnamon sugar while the butter is still moist enough to hold the coating.
Serve after a short cooling period. Allow unused donuts to cool completely before storing, keeping the sugary surfaces from rubbing against one another.
