Preheat oven to 350˚F and line a 9x13" pan with parchment paper.
Using two sheets of phyllo dough at a time, fold them accordion style, then arrange in the baking dish. Arrange the apple slices evenly throughout the folds.
Bake for 10 mins, then drizzle evenly with melted butter, making sure it goes in all of the nooks and crannies. Bake for another 10 mins.
Meanwhile, add milk, cinnamon, nutmeg and cloves to a small saucepan over medium low heat and whisk well to incorporate the spices. Let infuse for 5 minutes, then strain the milk into a bowl.
Add in the eggs, sugar, vanilla and salt and whisk well. Pour over the cake, making sure it gets between the folds. Return to oven for 35-40 mins, until golden brown.
For the syrup, combine the sugar, water and cinnamon sticks in a small saucepan. Bring to a boil, then reduce heat and simmer for 10 mins. Remove cinnamon sticks.
Drizzle the warm cinnamon syrup over the top of the baked cake.
