Preheat the oven to 350 degrees F. Grease a 9-inch pie dish, springform pan, or square baking pan.
In a medium bowl, combine the flour, baking powder, baking soda, and salt. Stir together and set aside.
In a large bowl, beat the eggs and sugar together for about 15-20 seconds with a hand mixer. Add the applesauce, Greek yogurt, oil, milk, and vanilla extract. Mix until smooth.
Add the dry ingredients into the wet and stir until just combined. There will still be some lumps - that is completely fine! Fold in the 1 cup of grated apple.
Pour half of the batter into the prepared pan and spread it out evenly. Lay the thinly sliced apples over the batter in about one or two layers.
Mix the cinnamon sugar together and sprinkle it evenly over the sliced apples.
Pour the remaining batter over the sliced apples and gently spread it out to cover.
To make the crumb topping, combine the flour, brown sugar, cinnamon, nutmeg, and a dash of salt in a medium bowl. Stir together. Add the cold butter pieces and use a pastry cutter or fork to cut the butter into the flour mixture until crumbly.
Sprinkle the crumb topping evenly over the batter. Bake for 35-45 minutes, or until the coffee cake is fully set and a toothpick inserted in the center comes out clean.
Let cool slightly before slicing and serving.