Preheat oven to 350F. Allow the butter and egg to reach room temperature.
Cream the butter with the brown sugar and granulated sugar. Add the eggs one at a time, mixing between additions. Add the vanilla extract.
In a separate bowl, whisk together the flour, salt, baking powder, nutmeg, cinnamon, and allspice.
Add the flour mixture to the wet ingredients. Mix until incorporated. While mixing, add the apple cider.
Bake at 350F for 12-15 minutes, until doughnuts are light brown in appearance. Allow doughnuts to cool.
For the topping, mix together the granulated sugar and cinnamon. Brush the doughnuts with the melted butter, and dip each one into the cinnamon-sugar mixture.