How to Make It
Preheat oven to 350°. Grease and flour a 12-cup bundt pan.
Cut the apples into pieces; shred using a food processor fitted with a grater attachment. Whisk together the flour, baking powder, salt, and cinnamon in a medium bowl. In a separate large bowl, whisk together the melted butter, sugars, milk, 1/4 cup apple juice, vanilla paste, and eggs. Gently fold the flour mixture into the butter mixture until just combined. Fold in the grated apple. Pour mixture into prepared pan.
Bake at 350° for 60 to 65 minutes, or until a skewer inserted into center of the cake comes out clean. While cake is baking, combine remaining 1/2 cup apple juice and apple liqueur in a small saucepan over medium-high heat. Simmer mixture, stirring occasionally, until reduced by about 1/4. Set aside 3 tablespoons of the apple juice mixture
Place cake in pan on a wire rack. Pierce the surface of the cake in pan with a skewer; brush cake with the remaining apple juice reduction. Let stand 15-20 minutes. Pour the reserved 3 tablespoons of apple juice reduction over cream cheese in a small bowl; whisk until smooth. Whisk in the confectioners’ sugar until smooth. Invert cake onto a serving plate and allow to cool slightly (15-20 minutes); drizzle with cream cheese mixture.