Place the sugar in a medium-sized bowl. With a citrus zester, zest the three lemons over the sugar, then mix the sugar and zest together.
Using a stand mixing or a hand-held mixer, add the softened butter to a large bowl and beat until light and fluffy on medium speed.
Gradually add the lemon-sugar mixture and mix on medium speed until incorporated.
With the mixer on medium-low speed mix in the cake flour ½ cup at a time, mixing well with each addition
Divide the dough in half and form into two logs ½" thick.
If you are using the sanding sugar, immediately roll the dough logs into the sugar to completely coat the dough. You want the dough to be a bit tacky from rolling into the logs for the sugar to stick.
Wrap the dough logs in plastic wrap and place in the refrigerator for one hour.
Preheat the oven to 300° F. Prepare a baking sheet with parchment paper or a silicone mat.
Slice the dough into ¼" slices and place on the baking sheets.
Bake the shortbreads for 30 minutes, let sit for 5 minutes on the baking sheet then place on a cooling rack to completely cool.
Store cookies in an airtight container for up to a week.