Cream together the butter, lemon juice, salt, and sugar until light and fluffy.
Add in the flour and mix until just combined. Using your hands, bring the mixture together to form a dough.
Wrap in plastic wrap and refrigerate for around 30 minutes to an hour.
Preheat the oven to 350 F. Line your baking sheet with parchment paper or a silicone mat.
Remove from the fridge, and roll out the dough on a floured surface to around 1/8 to 1/4 inches thickness. Using a cookie cutter (you can use whichever one you like), cut out your shortbread shapes. Place on a parchment lined baking sheet and chill for 30 more minutes.
Place in the oven and bake for 15 - 17 minutes, or until the edges are slightly golden. Remove from the oven and let cool completely.
Combine powdered sugar and lemon juice in a medium bowl and whisk. Add more lemon juice if needed. The glaze should be similar to the consistency of honey, a bit thick.
Drizzle or spread some lemon icing over the cookies, and add fresh lemon zest if desired.