For the Cake

1. Adjust oven rack to middle position and preheat oven to 350°F (180°C).
2. Grease a large (10- to 12-cup capacity) Bundt pan with 1 tablespoon plant-based butter.
3. In a medium saucepan, bring chopped apple and cider to a boil over medium-high heat. Reduce heat to medium and simmer until most of the cider has been absorbed and apples are easily smashed with a fork, 10 to 15 minutes.
4. Remove pan from heat, cool 5 minutes, then process  in food processor until puréed, 60 to 90 seconds, stopping to scrape down the sides as needed.
5. Measure out 1 cup apple mixture (if you have less than 1 cup apple mixture, add additional milk to reach 1 cup). Stir apple mixture into milk, set aside.
6. In a medium bowl, whisk together flour, baking powder, baking soda, salt, nutmeg, and mace; set aside.
7. In the bowl of a stand mixer fitted with the paddle attachment, beat remaining 8 tablespoons plant based butter, sugar, and brown sugar on medium speed until light and fluffy, about 3 minutes.
8. Add eggs, one at a time, beating well after each addition, about 20 seconds.
9. Add oil and beat until incorporated, about 45 seconds.
10. Decrease mixer speed to low and add flour mixture in 3 additions, alternating with apple mixture, scraping down sides and bottom of bowl with a rubber spatula as needed. Increase speed to medium and beat mixture just until combined, about 20 seconds.
11. Add vanilla and beat once more, just to combine, about 10 seconds.
12. Scrape batter into prepared pan, smooth surface with a spatula. Bake in preheated oven until a cake tester inserted in cake comes out with just a few crumbs, 45 to 50 minutes. Transfer cake to wire cooling rack set inside baking sheet and cool in pan 15 minutes, then invert directly onto wire cooling rack (see notes).

For the Sauteed Apples

1. Combine the apples and cider in a large skillet.  Sauté for around 10-15 minutes on low heat until the cider is mostly absorbed and the apples are caramelized.
2. Store in an airtight container and serve hot or cold.

For the Cinnamon Sugar Coating

1. Combine sugar, cinnamon, nutmeg, and salt in a small bowl. While cake is still hot, sprinkle with cinnamon sugar, using fingers to rub it onto sides.
2. Cool cake completely, about 1 hour, then serve topped with sauteed apples.