Preheat the oven to 350°F. Grease a 9-inch round baking pan.
Prepare the streusel by combining the flour, sugar, softened butter, vanilla and cinnamon, and nutmeg in a mixing bowl. Use a wooden spoon, rubber spatula, or your fingers to crumble the mixture into small lumps. Set aside.
In a medium mixing bowl, whisk together the flour, baking powder, cinnamon, and salt. Set aside.
In the bowl of a stand mixer with the paddle attachment or in a large mixing bowl with an electric hand whisk, beat the softened butter and sugar until it is pale and fluffy, after 3 to 5 minutes.
Add the egg and vanilla and beat the mixture again on a medium to high speed.
Add the milk and mix for 30 seconds.
Add the flour mixture to the butter mixture. Mix on a low speed until you no longer see dry pockets of flour. Scrape the sides and bottom of the bowl with a rubber spatula or spoon to ensure everything is well mixed.
Spread two-thirds of the cake batter into the prepared baking pan in an even layer. Spread the applesauce on top, and use the back of a spoon to spread it in an even layer. Spread the remaining cake batter on top of the applesauce in an even layer. Sprinkle the streusel topping on top.
Bake for about 40 to 50 minutes, until a toothpick or knife inserted into the center of the cake comes out clean. Allow to cool on a cooling rack to room temperature. Enjoy!!