Prepare the apple filling first and allow time for it to cool down.
Peel, core, and chop the apples.
Place the apples in a saucepan with the sugar, cinnamon, nutmeg, ground cloves, and allspice.
Cook the apples and spices over medium high heat, stirring occasionally. Reduce the heat to medium if the juices begin to bubble vigorously.
The apples are ready when they can be easily pierced with a fork, about 15 minutes. NOTE: Do not let the apples become too soft or you will end up with the consistency of applesauce. You wants chunks of cooked apples in this cake.
Place the cooked apples into a mixing bowl and set aside to cool.
Preheat the oven to 350 degrees F.
Prepare the springform pan. Remove the bottom of the pan and trace its shape on a sheet of parchment paper. Make sure you write on the side that will not touch the cake. Set aside.
Return the bottom to the springform pan and close. Spray the bottom and the sides of the pan with non-stick cooking spray.
Place the parchment paper circle in the bottom of the pan and lightly stray the paper with the cooking spray as well.
Sift the flour into a medium bowl.
Gently whisk the baking powder, baking soda, and cinnamon into the sifted flour. Set aside.
Add the butter and sugar to the mixing bowl of a stand mixer fitted with a paddle attachment and beat at low speed until fully combined, 1-2 minutes.
Add the eggs and vanilla to the butter and sugar mixture and beat together until combined.
Next place the Greek yogurt and milk into the bowl with the butter and sugar mixture. Beat at low speed until combined.
Add the dry ingredients to the wet ingredients and beat until just combined. Scrape down the sides and up from the bottom with a rubber spatula to be sure everything gets incorporated. For best results, never over mix cake batter. Set aside.
Using a slotted spoon, drain the cooked apple mixture and place it into a small bowl. You don't want to add too much liquid to your cake batter.
Add two tablespoons of flour to the apple mixture and stir to combine. This will help the apples stay distributed in the cake as it bakes.
Add the apple mixture to the batter, folding it in with a rubber spatula. Do not over mix.
Prepare the crumb topping by mixing the flour, brown sugar, and cinnamon in a small bowl.
Cut the butter into tablespoons and then quarter each tablespoon.
Using a pastry cutter or two forks, mix the butter into the flour and sugar mixture. The goal here is to create crumbs, so you are not trying to incorporate it completely. I usually finish up by rubbing the pieces of butter and the flour and sugar between my fingers. You do not want tiny crumbs or big chunks of butter; try for something in between.
Pour the cake batter into the prepared pan and smooth the surface with an offset spatula or the back of a spoon.
Sprinkle the crumbs evenly over the top of the batter. Press them down very slightly.
Bake until golden brown and a wooden skewer inserted into the center of the cake comes out clean, 50-60 minutes.
Allow the cake to cool for 15 minutes, then run a knife or a metal spatula around the exterior of the cake to make sure it is separated from the sides of the pan.
Open the springform pan and, using oven mitts or potholders, move the cake - on the base - to a wire rack to cool further.
Best Apple Coffee Cake with Cinnamon Crumb Topping is great served slightly warm.