Make the sugar cookie dough.
Using 1/4 of the batch, knead in the lemon zest and bake the cookies as instructed.
While the cookies cool, simply mix the lemon juice and erythritol together to make your icing. Note that you will want to adjust either the lemon juice or erythritol to get a slightly paste-like consistency. It should be thick but still have a bit of movement to it to make it easy to spread over the cookies. This can take a bit of guess work with the amounts, so you may have to play around with this a bit.
Spoon the icing over the cookies and place the entire cookie sheet in the fridge or freezer to chill the icing. Once hardened, you can stack these cookies for storage.
Store in the fridge or freezer.