Preheat oven to 350ºF. Prepare a large bundt pan by coating with butter and flour. Set aside.
Add chopped apple and apple cider in a small saucepan over medium-high heat. Bring to a boil, then reduce to a simmer and cook until the apples have softened and the cider has reduced, about 10 minutes. Remove pan from heat.
Use a fork to mash the apples, leaving no large chunks. Stir in milk, oil, and vanilla extract. Set aside.
In a small bowl whisk together the flour, baking powder, baking soda, salt, cinnamon, ginger, and nutmeg. Set aside.
In a large bowl, add butter and beat until creamy, using a stand mixer or hand mixer. Add sugar and brown sugar, and beat until mixture is fluffy, scraping down the sides of bowl as needed. Add eggs one at a time, beating well after each addition.
Alternate adding the dry ingredients and apple cider mixture in 3 parts, beginning and ending with dry ingredients, beating well after each addition.
Add the batter to the prepared bundt pan. Bake for 40-45 minutes, until a cake tester inserted in center comes out clean. Let the cake cool in pan for 15 minutes, then turn out onto a cooling rack to cool completely.
Brown Butter Cinnamon Sugar Topping
While the cake is cooling, make the brown butter. In a small saucepan, add butter and cook over medium-high heat. Stir frequently as the butter bubbles. After 2-3 minutes brown bits will appear on the bottom of the pan. Remove from heat and continue to whisk for 30 more seconds.
In a small bowl, stir together sugar, cinnamon, nutmeg, and salt. Brush the cake all over with brown butter, then immediately sprinkle and press the cinnamon sugar mixture all over the cake. Serve and enjoy!