Preheat the oven to 350 degrees F. Butter a 13-by-9-inch baking pan. Line with parchment paper, leaving a slight overhang.
In a medium bowl, whisk together the flour, cinnamon, baking powder and salt. In a large bowl, beat the sugars and butter with an electric mixer on medium until light and fluffy, about 3 minutes. Add the eggs and vanilla and beat to combine. Add the flour mixture to the butter mixture and beat to combine. Fold in the apples and walnuts.
Transfer the batter to the prepared pan and spread it out into an even layer. Bake until a toothpick inserted into the center comes out with moist crumbs attached, 20 to 25 minutes. Transfer the pan to a rack to cool completely. Then lift the bars out using the parchment overhang and transfer to a cutting board. Cut into 24 pieces.