Preheat the oven to 350 degrees F. Spray a 10-cup bundt pan with nonstick baking spray.
Place the chopped apple and apple cider in a saucepan and heat over medium-high heat. Brind the mixture to a boil, then reduce it to a simmer and cook until the apple is soft and the cider is absorbed. Use a fork to mash the apple as much as possible. You don’t want any large chunks! Measure out 1 cup of the apple mixture. Stir in the oil, milk and vanilla.
In a small bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, nutmeg and allspice.
In the bowl of an electric mixer, beat the butter until creamy. Beat in the sugars until the mixture is fluffy, about 5 minutes, scraping down the bowl as needed. Beat in each egg one at a time until fully combined.
Alternate adding the dry ingredients and the apple cider/milk mixture, beginning and ending with the flour. I like to do this in three increments, so add a fourth of the dry ingredients, then a third of the liquid, and so on until you finish with the dry ingredients, beating until just combined.
Spread the batter into the greased bundt pan, smoothing out the top with a spatula. Bake the cake for 35 to 45 minutes, or until a tested inserted in the cake comes out clean. Let the cake cool in the pan for 15 minutes, then gently removed the cake from the pan. Let the cake cool completely.
To brown the butter, place butter in a small saucepan over medium heat and immediately begin whisking. Stir as it bubbles, and after 2 to 3 minutes or so you should see brown bits appear on the bottom of the pan. Remove from heat immediately and continue to whisk for another 30 seconds.
In a small bowl, stir together the sugar, cinnamon, nutmeg and salt.
Brush the cake all over with the brown butter. Sprinkle and press the cinnamon sugar into the cake right after brushing with the brown butter. You can rub the sugar onto the sides and the center so it sticks. Serve as desired!