Preheat oven to 350°F. Butter and flour a 9x13" baking dish or spray with non-stick baking spray.
Mix flour, butter, sugar, baking powder, salt, egg, and vanilla together in a large bowl using an electric mixer until combined.
2 cups all-purpose flour, 3/4 cup salted butter, 1/2 cup granulated sugar, 1/2 teaspoon baking powder, 1/4 teaspoon salt, 1 large egg, 1 teaspoon pure vanilla extract
Pat the cake dough into an even layer on the bottom of the prepared baking dish using the back of a rubber spatula or your hands. Refrigerate until apple filling is ready.
In another large bowl, combine the apples, lemon juice, lemon zest, sugar, cinnamon, and applesauce. Spoon apple filling in an even layer over the cake dough.
5 apples, 1 1/2 cups unsweetened applesauce, 2 Tablespoons granulated sugar, 1 Tablespoon freshly squeezed lemon juice, 1 teaspoon lemon zest, 1/2 teaspoon ground cinnamon
In another large bowl, cut the butter into the flour and sugar using a pastry cutter until it resembles coarse crumbs. Squeeze handfuls of streusel topping together to create larger clumps, then sprinkle the streusel over the apple filling.
2 cups all-purpose flour, 3/4 cup cold salted butter, 3/4 cup granulated sugar
Bake for 45 minutes until lightly golden brown on top and the apples are soft.
Cool completely, then cut into squares and serve with sweetened whipped cream or vanilla ice cream. I think this cake tastes best cold, straight from the fridge.