Line an 8” square pan with parchment paper so the paper hangs over the sides.
Cream the butter and sugar for 3-5 minutes, until light and fluffy.
Beat in the lemon zest and vanilla, then add in the flour and salt and mix until just combined.
Place the dough into the prepared pan and press the dough evenly into a pan. Using a flat bottomed glass helps to get it packed tightly and evenly. It should be about 1/2" thick.
Use a toothpick or skewer to poke holes into the shortbread. You could also use a fork.
Bake for 12-15 minutes, until lightly golden brown.
Allow to cool in the pan for 10 minutes, and then lift the shortbread out of the pan by the parchment paper overhang.
Place the shortbread on the parchment paper onto a cutting board and use a long sharp knife to cut the shortbread into squares that are 1" wide.
Allow to cool all the way before drizzling with lemon icing.
Lemon Icing
In a small bowl, combine the powdered sguar and enough lemon juice to make a drizzly icing. Adjust the level of lemon juice depending on your preferences - more lemon juice will make a thinner drizzle and less will make it thicker. Drizzle the icing over the cooled shortbread cookies.
Notes
Store these cookies in a metal or glass container with a lid on the counter for up to 5 days.