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.: As the oven comes up to 350°F , you should feel that warm anticipation in the kitchen and notice the dry, warm smell as the heating elements cycle on. Rubbing a fingertip near the oven rack will reveal increasing warmth; this steady heat ensures an even rise and a golden top. Lining the pan with parchment creates handles that make removing the cake easy, and the baking spray with flour prevents sticking while contributing to a clean release. One common mistake is skipping the parchment which can make release difficult and damage the cake edges. Using a pan that is too large or too small will change bake times, so use the recommended 8 inch square pan for consistent results. If your oven runs hot, consider reducing the temperature by ten degrees to avoid an over browned exterior before the center sets.
In a medium bowl, whisk together flour, baking powder, cinnamon, and salt. Add 1 1/2 cups (174 grams) apples, tossing to coat.: At this stage you will notice the dry mixture takes on the faint sweet perfume of cinnamon . Whisking aerates the all purpose flour and evenly distributes the baking powder so pockets of rise are consistent. Tossing the diced apples in the dry mix helps suspend them through the batter and prevents sinking to the bottom. If the apples are too wet they may clump, so patting them dry first is useful. A frequent error here is over mixing the flour into a paste if you add the wet ingredients while still vigorously stirring, which can tighten the crumb and make the cake dense.
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 1/2 cup (58 grams) apples.: As you whisk the brown sugar with the egg and egg white , the bowl should lighten in color and take on a slightly fluffy texture, which traps air to help the cake rise. When you add the melted unsalted butter , the mixture smells richer, and the tang from the sour cream cuts through the sweetness, giving depth. Folding in the dry mix gently preserves the trapped air, ensuring a tender crumb. The batter may show streaks, this is okay; over mixing will develop gluten and reduce tenderness. Smoothing the top with a spatula gives a uniform bake surface, and scattering the final diced apples on top creates appealing fruit pockets and a slightly caramelized surface. Watch the consistency, if the batter seems overly stiff you may have over measured the flour . A common misstep is vigorously mixing here, which flattens the batter and yields a tougher cake.
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.: As the cake bakes you will see the top turn a warm golden brown and the kitchen will fill with a toasty, sweet aroma of apples and cinnamon . The edges will pull slightly from the pan sides, signaling doneness. The wooden pick should emerge free of wet batter, a few moist crumbs are fine. Allowing the cake to rest in the pan for ten minutes helps it set, reducing the risk of breaking when you lift it out. Cooling on a wire rack prevents steam from pooling under the cake, which could sog the bottom. A common error is cutting too soon, which makes slices fall apart. If the center seems underdone but the top is browning, tent loosely with foil and continue baking in five minute increments to avoid over browning.
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.: The glaze should be silky and pourable, offering a sweet sheen that settles into nooks and highlights the golden top. Whisk the confectioners sugar and whole milk until no lumps remain, and adjust the thickness by adding tiny amounts of milk if needed. Drizzling while the cake is still warm helps the glaze set with a delicate crust and soak slightly into the surface for a pretty finish. Beware of making the glaze too thin which will run off the cake; too thick and it will look clumpy. Serve immediately for the best contrast between warm crumb and cool glaze, or let the cake sit for a few minutes if you prefer a firmer topping.