Preheat oven to 350 degrees F. Lightly grease the bottoms of three 8-inch round cake pans. Line bottoms with waxed paper; grease and lightly flour pans.
In a large nonstick skillet melt the 3 tablespoons butter. Add apples, apple cider, and Calvados. Cook over medium-high heat for 6 to 8 minutes or until liquid has reduced by half, stirring occasionally.
In a large bowl stir together flour, sugar, baking powder, cinnamon, nutmeg, baking soda, salt, and cloves. In another bowl stir together eggs, the 2/3 cup melted butter, and the apple mixture. Stir egg mixture into flour mixture. Spread batter into prepared pans.
Bake 35 to 40 minutes or until a toothpick comes out clean. Cool layers in pans for 10 minutes. Remove layers from pans; peel off waxed paper. Cool on wire racks.
Meanwhile, prepare Apple Cider Glaze. To assemble, using a serrated knife, cut each cake layer in half horizontally. Evenly brush Apple Cider Glaze over layers; let stand 10 minutes. Prepare Butterscotch Frosting. Place one cake layer, glazed side up, on a serving plate. Spread with 1/2 cup Butterscotch Frosting. Repeat with four more layers and frosting. Top with remaining layer, top side up, and frost top and sides. Cover and chill cake to store. If desired, just before serving, microwave remaining Butterscotch Sauce for 20 to 30 seconds or until warm and drizzling consistency, stirring once. Drizzle over top edge of cake, letting sauce drip down sides.
Apple Cider Glaze
In a small saucepan melt butter. Stir in dark brown sugar, apple cider, Calvados, and heavy cream. Bring to boiling; reduce heat. Simmer, uncovered, for 3 minutes. Remove from heat and cool.
In a large bowl beat cream cheese, butter, and vanilla with a mixer on medium speed about 1 minute or until light and fluffy. Gradually beat in powdered sugar to reach a thick but spreadable consistency. Beat in Butterscotch Sauce. Makes 5 1/2 cups.
In a medium heavy saucepan, melt butter over low heat, stirring often. Increase heat to medium. Stir in light-colored corn syrup, granulated sugar, dark brown sugar, water, and salt. Bring to boiling, stirring constantly; reduce heat. Simmer, uncovered, for 5 minutes, stirring often. Meanwhile in a small bowl whisk together whipping cream, cornstarch, and vanilla. Carefully stir in whipping cream mixture. Return to boiling. Cook and stir until thickened and bubbly. Cook and stir for 1 minute more. Transfer sauce to an airtight container and completely chill before using for Butterscotch Frosting and Applicious Cider Cake with Butterscotch Frosting. Makes 1 1/2 cups