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Preheat the oven to 375 degrees Fahrenheit. Butter an 8-inch square pan or equivalent-size baking dish.
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In a small bowl, whisk together the flour, baking powder, and salt. In a medium bowl, cream together the butter and 1 cup of the sugar. Stir in the eggs and vanilla. Add the flour mixture and beat until combined. Spread the mixture evenly in the prepared pan.
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In a small bowl, combine the remaining 1/3-cup sugar with the cinnamon. Squeeze lemon juice into a medium bowl. Peel, core, and slice the apples into the bowl. Add the cinnamon-sugar mixture and toss to thoroughly coat each apple slice. Arrange the apple slices on top of the batter in overlapping rows, pressing lightly into the batter. Bake for 45 minutes, until a cake tester or toothpick inserted in the center comes out clean. Cool in the pan on a wire rack for 10 to 15 minutes before serving.
