Preheat your oven to 350°F (175°C). Grease and flour 2 6-inch round cake pans (or 3 4-inch cake pans).
In a large bowl, whisk together the flour, baking powder, baking soda, salt, and cinnamon until well combined. Set aside.
In a medium bowl, combine the applesauce, Greek yogurt, and milk. Whisk till smooth and set aside.
In a large mixing bowl, cream together the softened butter and granulated sugar until light and fluffy.
Beat in the eggs, one at a time, until well incorporated. Stir in the vanilla extract.
Gradually add the dry ingredients to butter and egg mixture, alternating with the applesauce mixture, beginning and ending with the dry ingredients. Mix until just combined.
Pour the batter into the prepared cake pans and smooth the top with a spatula.
Bake in the preheated oven for 25-35 minutes, or until a toothpick inserted into the center of the cake comes out clean. Smaller cakes will bake a little faster than larger cakes so watch them closely and check for doneness beginning at 20 minutes.
Allow the cakes to cool in the pans for 10 minutes before transferring them to a wire rack to cool completely.