Preheat the oven to 350°F. Grease an 8-inch square baking dish.
In a medium bowl, combine the flour, powdered sugar, and salt. Cut in the softened butter until the mixture resembles coarse crumbs. Press this mixture evenly into the bottom of the prepared baking dish.
Bake the crust for 15 minutes or until lightly golden. Remove from the oven and let it cool slightly.
In another bowl, whisk together the eggs, granulated sugar, lemon juice, lemon zest, baking powder, and vanilla extract until smooth. Pour this mixture over the baked crust.
Return to the oven and bake for an additional 20-25 minutes, or until the filling is set and lightly golden on top. Remove from the oven and let it cool completely in the pan.
Once cooled, dust with powdered sugar, cut into squares, and serve.