Heat oven to 350 degrees. Brush an 8-inch-square baking dish with oil, then line with parchment, leaving an overhang, and brush parchment with oil. Set aside.
Make the crust: Stir together 1/2 cup oil, sugar, vanilla, and salt. Add flour and stir until just combined. Press dough into prepared dish and bake until pale golden-brown, about 20 minutes.
Meanwhile, make the filling: Process tofu, sugar, lemon zest and juice, baking powder, and flour in a food processor until smooth. Pour onto baked crust and bake until set, 25 to 30 minutes more. Let cool completely, then dust with confectioners' sugar and cut into squares.
They'll Never Guess
Yes, tofu even does dessert: Combined with sugar, lemons, coconut oil, and whole wheat, it provides the basis for an airy, ethereal lemon square.