Preheat oven to 350°F. Grease a 4-inch round cake pan and line bottom with parchment.
Whisk flour, baking powder, salt, 1/4 tsp cinnamon, and brown sugar.
Mix apple purée, egg yolk, melted butter, milk, and vanilla. Fold into dry ingredients until just combined.
Pour batter into pan. Smooth top.
Mix granulated sugar with 1/2 tsp cinnamon. Sprinkle evenly over batter.
Bake 18–22 minutes until toothpick comes out with a few crumbs.
Cool 15 minutes, remove from pan, cool fully on rack.
Spread thin layer of soft cream cheese frosting. Chill 10 minutes so crust stays crisp.