Peel and cut the apples into ½-inch cubes, then toss them with sugar and cinnamon until evenly coated. Set them aside so they can release a little juice and develop more flavor.
Preheat the oven to 350°F (175°C) and line an 8-inch (20 cm) springform pan with parchment paper. In one bowl, whisk together the yogurt, sugar, vanilla extract, and melted butter until smooth.
In another bowl, combine the flour, baking powder, baking soda, cinnamon, and salt. Gently mix the dry mixture into the wet mixture until just combined. The batter will be thick, which helps hold the apple filling in place.
Spread half of the batter evenly into the prepared pan, then layer all of the cinnamon apples on top, don't use excess liquid released from apples. Spoon the remaining batter over the apples and gently spread in an even layer to cover. Sprinkle all of the chilled crumble topping generously over the surface.
Bake the cake for 40–50 minutes, or until the top is golden brown and a toothpick inserted in the center comes out mostly clean. Let the cake cool completely on a wire rack before adding the icing.