Preheat the oven to 350°F and grease a 9x13-inch baking pan. In a large bowl, beat the softened butter and brown sugar until light and fluffy.
Mix in the eggs, applesauce, milk, and vanilla extract until smooth and fully combined.
Add the flour, baking powder, cinnamon, and salt. Mix just until combined, then spread the batter evenly into the prepared baking pan.
In a medium bowl, ideally using your hands, combine the flour, brown sugar, cinnamon, and melted butter until coarse crumbs form.
Sprinkle the crumb topping evenly over the batter. Bake for 38 to 42 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cake to cool for about 20 minutes in the pan.
Whisk together the powdered sugar, milk, and vanilla extract until smooth, then drizzle over the warm or cool cake before slicing and serving.