Cook the apples, sugars, spices, cider, and lemon juice in a pan over medium heat until soft and thick, about 20-25 minutes. Let the filling cool.
Preheat the oven to 350°F (175°C). Grease and flour five 9-inch round cake pans.
Mix the dry ingredients—flour, baking soda, baking powder, salt, cinnamon, nutmeg, and cloves—in one bowl.
In another bowl, cream butter and sugar together until fluffy. Add eggs one at a time, then stir in vanilla.
Alternate adding the dry mix and buttermilk to the butter mixture until combined.
Divide the batter evenly among the pans.
Bake for 20-25 minutes, until a toothpick comes out clean. Cool the cakes on wire racks.
Place one cake layer on a plate and spread apple filling on top. Repeat with the remaining layers.
Mix powdered sugar, apple cider, and vanilla until smooth.
Pour the glaze over the cake, letting it drip down the sides.
Let the cake sit for 30 minutes before serving.