Preheat oven to 350°F (180°C). Spray an 8-inch square baking pan with baking spray with flour. Line pan with parchment paper, letting excess extend over sides of pan.

In a medium bowl, whisk together flour, baking powder, cinnamon, and salt. Add 1½ cups (174 grams) apples, tossing to coat.

In another medium bowl, whisk together brown sugar, egg, and egg white until light and fluffy, about 2 minutes. Whisk in melted butter, sour cream, applesauce, lemon juice, and vanilla until well combined. Using a rubber spatula, fold in flour mixture until just combined and no dry streaks remain. (Batter will not be completely smooth.) Pour batter into prepared pan, and smooth top. Top with remaining ½ cup (58 grams) apples.

Bake until golden brown and a wooden pick inserted in center comes out clean, 40 to 45 minutes. Let cool in pan for 10 minutes. Using excess parchment as handles, remove from pan, and let cool on a wire rack for 10 minutes.

In a small bowl, whisk together confectioners’ sugar and milk until smooth; drizzle onto warm cake. Serve immediately. Store in an airtight container for up to 3 days.