How to Make It
MAKE THE CAKE Preheat the oven to 350°. Butter and flour a 9-inch springform pan and line the bottom with parchment.
In a small saucepan, bring the apple cider to a simmer and cook over moderate heat until reduced to 1/4 cup, about 12 minutes. Remove the pan from the heat and let cool slightly.
In a large skillet, melt 2 tablespoons of the butter. Add the apples, the granulated sugar and 1/4 teaspoon of salt and cook over moderately high heat, stirring occasionally, until the apples are just tender, about 3 minutes. Remove the skillet from the heat and let the apples cool slightly.
In a medium bowl, whisk 1 1/2 cups of the all-purpose flour with the whole-wheat flour, baking powder, cinnamon, ginger, nutmeg and 1 1/4 teaspoons of salt.
In a stand mixer fitted with the paddle, beat the remaining 2 sticks of butter with the brown sugar at medium speed until light and fluffy, about 3 minutes. Beat in the eggs 1 at time, mixing well after each addition and scraping down the side of the bowl as needed. (The batter will look broken.) Beat in the cooled reduced apple cider and vanilla until just combined. At low speed, beat in the dry ingredients until just combined. Using a rubber spatula, fold in the cooked apples.
Scrape the batter into the prepared pan, and using a spatula, smooth the top. Bake for about 50 minutes, until a toothpick inserted into the center of the cake comes out clean. Transfer the cake to a wire rack set in a rimmed baking sheet and let cool slightly, about 20 minutes. Invert the cake onto a large plate and remove the parchment paper. Invert the cake back onto the wire cooling rack and let cool completely.
MAKE THE GLAZE In a small saucepan, bring the apple cider to a simmer and cook over moderate heat until reduced to ¼ cup, about 12 minutes. Remove the pan from the heat and whisk in the butter until it melts. Whisk in the confectioners’ sugar and salt until smooth. Let cool slightly, about 15 minutes. Pour the syrup over the cake and serve.