Step 1: In a medium bowl, combine the diced apples with 1 tablespoon of granulated sugar and 1 teaspoon of ground cinnamon. Toss to coat evenly and set aside.
Step 2: In a small bowl, whisk together the ½ cup all-purpose flour, ¼ cup granulated sugar, ¼ cup light brown sugar, and ¼ teaspoon ground cinnamon.
Step 3: Add the cold, cubed butter to the flour mixture. Using your fingertips or a pastry blender, cut the butter into the dry ingredients until coarse crumbs form. Avoid overmixing. Place the bowl in the refrigerator while you prepare the cake batter.
Step 4: Preheat your oven to 350°F (175°C). Grease and flour an 8x8 inch baking pan or line with parchment paper.
Step 5: In a medium bowl, whisk together 1 ½ cups all-purpose flour, 1 teaspoon baking powder, and ½ teaspoon salt. Set aside.
Step 6: In a large bowl, cream together the softened ½ cup unsalted butter and ¾ cup granulated sugar with an electric mixer until light and fluffy.
Step 7: Beat in the large egg until well combined, then stir in the 1 teaspoon vanilla extract.
Step 8: Gradually add the dry ingredients to the wet ingredients, alternating with the ½ cup milk, beginning and ending with the dry ingredients. Mix until just combined; do not overmix.
Step 9: Pour the cake batter into the prepared baking pan and spread evenly.
Step 10: Evenly scatter the spiced apples over the cake batter.
Step 11: Remove the crumb topping from the refrigerator and sprinkle it generously over the apples.
Step 12: Bake for 45-55 minutes, or until a toothpick inserted into the center of the cake comes out clean. If the topping starts to brown too quickly, you can lightly tent it with foil.
Step 13: Let the cake cool in the pan on a wire rack for at least 15-20 minutes before slicing and serving.