Preheat the oven to 350°F (175°C) and grease a 9-inch round cake pan.
Whisk together flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
Beat the softened butter, granulated sugar, and brown sugar until light and fluffy.
Add the eggs one at a time, mixing well after each addition.
Mix in the vanilla extract and applesauce.
Add the dry ingredients alternately with the apple cider, beginning and ending with the dry ingredients.
Fold the diced apple gently into the batter.
Pour the batter into the prepared cake pan and smooth the top.
Combine the cinnamon and sugar for the topping and sprinkle evenly over the batter.
Drizzle the melted butter over the cinnamon-sugar mixture.
Bake for 38–42 minutes, or until a toothpick inserted into the center comes out clean.
Cool the cake in the pan for 10–15 minutes before transferring it to a serving plate.
Slice and serve warm or at room temperature.