Preheat oven to 400 degrees. Grease the wells of the doughnut pan with some butter.
In a glass measuring cup, mix together milk and vinegar and allow to sit at room temperature for at least 5 minutes to curdle.
In a large bowl whisk together flours, baking powder, cinnamon, nutmeg and salt.
In a small bowl, whisk together melted butter, sugar, honey, egg, vanilla and curdled milk.
Whisk wet ingredients into dry ingredients, taking care not to overmix.
Spoon batter evenly into wells of the doughnut pan, filling only 2/3 full.* If filled too high, there won’t be any doughnut holes. Another way to fill the wells is by transferring the batter to a quart-size Ziploc bag and snipping one corner ½inch and piping the batter into the wells. Bake for 8-10 minutes or until doughnuts spring back when pressed lightly. Allow to cool 5 minutes before turning doughnuts onto a cooling rack.
While the doughnuts are still warm, combine the 2 Tablespoons cane sugar with the teaspoon of cinnamon in a shallow bowl. Brush the tops (or bottoms) of the doughnuts with melted butter and dip into the cinnamon–sugar mixture so that it sticks. Repeat with remaining doughnuts.