Preheat oven to 350°F (175°C).
Cream butter, white sugar, and brown sugar together until light and fluffy. Add eggs and vanilla extract, stirring until just combined. Mix in applesauce.
Sift flour, baking powder, cinnamon, nutmeg, allspice, and salt into the wet ingredients. Gradually whisk in the dry ingredients until fully incorporated.
Fold in diced apples.
Grease and flour a 9x5-inch loaf pan, or line with parchment paper.
Transfer half of the batter into the prepared pan. Smooth the top and tap the pan to remove air bubbles. Spread half of the cinnamon sugar mixture over the batter. Add the remaining batter on top.
Prepare crumb topping by mixing brown sugar, butter, flour, and a dash of cinnamon. Sprinkle over the batter. Distribute remaining diced apples on top.
Bake for 50 minutes to 1 hour, checking for doneness with a cake tester.
Allow to cool in the pan on a wire rack for 30 minutes to 1 hour before slicing.