Apple Rum Spice Cake

Ingredient:

• 1 medium McIntosh, Braeburn, or Fuji apple, peeled, cored, and cut into 1/2-inch pieces
• 1 medium Granny Smith apple, peeled, cored, and cut into 1/2-inch pieces
• 5 tablespoons packed light brown sugar
• 1 tablespoon all-purpose flour
• 1/4 teaspoon kosher salt
• 1/8 teaspoon ground cinnamon
• 1/2 cup (4 ounces) gold rum (such as Bacardí)
• 1 teaspoon fresh lemon juice
• 1/2 teaspoon vanilla extract
• Cake Layers
• 2 1/4 cups sifted bleached cake flour (such as Swans Down) (about 7 1/2 ounces), plus more for dusting pans
• 1 1/4 cups granulated sugar
• 1 tablespoon baking powder
• 1 teaspoon ground cinnamon
• 1/2 teaspoon freshly grated nutmeg
• 1/2 teaspoon kosher salt
• 5 large eggs, separated
• 2/3 cup water
• 1/2 cup canola oil
• 2 tablespoons fresh lemon juice
• 1 teaspoon vanilla extract
• 1/2 teaspoon cream of tartar
• Ginger and Cream Cheese Frosting
• 1 (4-inch) piece fresh ginger, unpeeled and grated on a Microplane (about 3 tablespoons)
• 8 ounces cream cheese, softened
• 1/3 cup granulated sugar
• 1 1/2 teaspoons vanilla extract
• 1/4 teaspoon kosher salt
• 2 cups heavy cream