Preheat oven to 350°F (177°C). Spray donut pans with non-stick spray. Set aside.
Whisk the flour, baking soda, baking powder, cinnamon, nutmeg, and salt together in a large bowl until blended. Set aside.
Whisk the melted butter, egg, brown sugar, granulated sugar, apple cider, apple butter and vanilla extract in a medium-sized bowl until combined.
Pour wet sugar mixture into the dry flour mixture. Whisk everything together until smooth and combined. The batter will be slightly thick.
Fill the donut pan. Fill each donut cup about halfway. An easy way to transfer the batter is to use a large gallon-sized zipped-top bag. Add the batter to the bag, cut off a corner, and pipe it into the donut pan filling it about halfway.
Bake at 350°F for 9-10 minutes. Once they are out of the oven, let them cool for 2 minutes, then invert the pan to release the donuts. You can then re-grease the donut pan and bake the remaining donut batter. If you have 2 donut pans, you can bake both trays at once.
Glaze the donuts. Combine the melted butter, apple butter, powdered sugar, vanilla, salt, and one tablespoon of apple cider together until well combined. If the glaze is thick add the second tablespoon of apple cider and mix until combined. Once cool enough to handle, dunk one side of each donut in the glaze. Place them on a wire rack set on top of parchment paper to catch any drips.