Preheat oven to 400 degrees F. Grease a 10-inch springform pan and set aside.
Add butter, sugar, eggs, vanilla, flour, baking powder, salt, and milk to a large mixing bowl or the bowl of a stand mixer. Use a hand mixer (or paddle attachment if using stand mixer) and beat on medium-low speed until all of the ingredients are incorporated. Fold in apples.
Transfer batter to prepared pan and bake for 20 minutes. While cake is in the oven prepare the topping but heating butter and sugar in a medium saucepan over medium heat. Remove from heat and let cool 10 minutes. Once slightly cooled, whisk in eggs.
After 20 minutes, remove cake from oven and pour topping evenly over cake. Return to oven and bake 20 more minutes. Sprinkle sliced almonds and bake for 3-5 minutes, or until almonds just begin to turn golden.
Remove cake from oven and let cool 15 minutes before removing from springform pan. Cake can be served warm or cold. To serve cake cold, let cool completely then chill in the refrigerator for at least 4 hours or overnight.
Store cake, covered, in the fridge for up to 3 days.