Preheat the oven to 350°F. Line an 8x8-inch baking pan with parchment paper.
Prepare the vegan egg replacer according to package directions. Set aside.
In a small bowl, sift together the flour, baking powder, salt, and cinnamon. Set aside.
In a large bowl, stir together the brown sugar, granulated sugar, melted butter, and vanilla extract just until combined. Stir in the prepared egg replacer.
Add the apples and walnuts then gradually stir in the flour mixture, just until fully incorporated.
Into the prepared baking pan, press the bar mixture in an even layer to cover the bottom of the pan.
Bake for 30 minutes, until bars are set and golden in color. Allow to cool completely then use the parchment paper to lift the bars out of the pan and onto a flat surface. Cut into squares and serve.