In a medium saucepan over medium-high heat, heat apple cider until reduced to 1/2 cup (usually takes about 12 – 15 minutes). Transfer to a heat-proof bowl and place into refrigerator to cool completely (about 30 minutes). 

In a large bowl, whisk together flour, brown sugar, baking powder, baking soda, salt, cinnamon and nutmeg. 

Remove cooled apple cider for refrigerator and add the egg and cooled butter. Whisk until smooth. It's important for the butter to be completely cooled, otherwise the mixture will curdle/egg will cook. Pour over the dry ingredients and use a rubber spatula to stir just until moist. Cover and place in refrigerator to chill for one hour. 

When ready to fry, heat the oil to 360F. I used a deep fryer, but you could also use a large stockpot or Dutch oven. Place a cooling rack on top of a baking sheet and cover with a paper towel (the baking sheet just makes cleanup easier), set aside. 

Working on a lightly floured surface, roll dough until it's 1/2-inch thick. Cut using a round cookie cutter about 1 3/4" in diameter. For the smaller hole, I used a piping tip about 1/2" in diameter. 

Place several doughnuts (I used 3 at a time) in the oil and cook for about 45 minutes on each side until golden brown and crispy. Transfer to covered cooling rack. Repeat until all doughnuts are fried. 

In the bowl of a standing mixer fitted with the paddle attachment (or in a large bowl using a handheld mixer), beat cream cheese and butter on high speed until smooth and creamy.

Add powdered sugar, vanilla, cinnamon, nutmeg and salt. Mix on low speed until combined, then turn up to high speed and beat for 1 – 2 more minutes. You can add more powdered sugar if you want the frosting to be thicker.

Transfer to piping bag fitted with desired attachment and place a dollop on top of each doughnut. It's best if the doughnuts have cooled slightly, otherwise the frosting will melt. Enjoy warm!