Preheat your oven to 350°F (175°C). Grease a 9-inch square baking pan.
In a bowl, whisk together flour, baking soda, salt, cinnamon, and nutmeg.
In another bowl, beat the butter and sugar until fluffy. Add the egg and applesauce, mixing well.
Gradually add the dry ingredients to the wet mixture, stirring until just combined.
Pour the batter into the prepared pan.
For the crumb topping, mix flour, brown sugar, and cinnamon in a bowl. Add melted butter and mix until crumbly.
Sprinkle the crumb topping evenly over the batter.
Bake for 45 minutes or until a toothpick inserted into the center comes out clean.
Let the cake cool in the pan before serving.