Heat oven to 325°F (165°C). Lightly coat an 8-inch square or 9-inch cake pan with butter or nonstick spray. For extra security, line it with parchment paper.
Toss apples with lemon juice, then cinnamon and sugar and set aside.
Whisk butter, sugars, cinnamon, and salt together until evenly mixed. Add flour and mix until it disappears. Set it aside.
Beat butter with sugar until lightened and fluffy. Add egg, sour cream, and vanilla and beat until combined. Sprinkle surface of batter with baking powder and salt, and beat well to combine. Add flour and mix only until it disappears.
Scrape batter into prepared cake pan and smooth it flat. Arrange apples on cake, slightly overlapped. Pour any cinnamon-apple juices in bottom of bowl over apples. Sprinkle crumbs over apple slices.
Bake the cake until a toothpick inserted into the apples doesn’t hit any crisp spots and juices are bubbling around some apples, about 50 to 55 minutes.
Cool to room temperature before cutting into squares or wedges. Dust generously with powdered sugar.
