Preheat the oven to 350°F and line an 8×8-inch pan with parchment paper.
Whisk flour, granulated sugar, brown sugar, salt, and baking powder in a bowl.
Cut in cold butter until the mixture resembles coarse crumbs.
Stir in egg and vanilla just until a soft crumb forms.
Press two thirds of the mixture firmly into the pan to form the crust.
Whisk eggs, sugar, flour, baking powder, lemon juice, and zest until smooth.
Pour filling over the crust and sprinkle remaining crumbs on top.
Bake for 35 to 40 minutes until lightly golden and set.
Cool completely, then lift out and slice into squares.