With a mixer, combine the granulated sugar, butter, olive oil and lemon zest . Add the eggs, one at a time, and continue to beat for 1 minute.
Whisk together the flour, baking powder and salt, and mix into the lemon mixture until a soft dough forms. Cover the dough with plastic wrap and refrigerate, 1 hour.
Preheat the oven to 350 degrees F, and line 2 cookie sheets with parchment paper or silicone baking mats.
Divide the dough into thirds. Cut each third in half, then repeat. (You should have 12 small pieces of dough.)
On a surface lightly dusted with flour, roll each piece out into a 1/2-inch-thick log, then cut each into 3 pieces. Coil each piece into a small bun shape, bringing the end up over the top and pinching to seal. (Don't worry if they look funny; the glaze will cover any screw-ups.) Repeat with all the pieces, then arrange on the prepared cookie sheets at least 3 inches apart.
Bake in batches, rotating the pans halfway through, 18 to 20 minutes. Set aside to cool completely, about 20 minutes.
Place the confectioners' sugar in a small bowl and break up any large clumps with a whisk. Add the juice of 1/2 a lemon and whisk until a smooth glaze forms (if the glaze is too thick, add a bit more lemon juice; if it is too thin, add a bit more sugar).
Dip the tops of each cookie into the glaze and allow the excess to drip back into the bowl. Place on a wire rack to dry, then store in an airtight container. Cookies will keep for 5 days at room temperature, or 3 weeks frozen.