Preheat the oven to 350°F. Grease an 8-inch square baking dish.
In a medium bowl, combine the flour, powdered sugar, and salt. Add the softened butter and mix until crumbly.
Press the mixture evenly into the bottom of the prepared baking dish to form the crust. Bake for 15 minutes or until lightly golden.
While the crust is baking, in a separate bowl, whisk together the eggs, granulated sugar, lemon juice, lemon zest, baking powder, and vanilla extract until smooth.
Pour the lemon mixture over the baked crust and return to the oven. Bake for an additional 20-25 minutes, or until the filling is set and slightly golden around the edges.
Remove from the oven and let cool completely in the baking dish on a wire rack. Once cooled, dust with powdered sugar and cut into squares.