In a large bowl, whisk together the flour, zest, baking soda and salt. In a liquid measuring cup, stir together the cream, lemon juice and vanilla.
In a stand mixer with the paddle attachment, cream the butter and sugar together on medium high speed for several minutes until light and fluffy.
Turn the speed down to low and add the eggs one at a time, allowing each egg to fully incorporate into the batter. Scrape down the side of the bowl.
With the mixer running on medium low, alternate between adding the flour and cream mixture (which will have slightly curdled from the lemon juice) until all of the ingredients are incorporated. Use a spatula to scrape down to the bottom of the bowl one last time and give the batter a final stir.
Use a 1-inch cookie scoop to potion the dough onto the parchment (note: these cookies spread while baking so leave room between each cookie. Bake in batches if you have smaller baking sheets). Allow the cookies to chill in the refrigerator for 25 minutes.
Preheat the oven to 350 degrees F. Whisk together the glaze ingredients and set aside.
Bake the cookies for 15-18 minutes, or until the edges are just starting to brown. Allow to cool for several minutes and then brush with the glaze.
Notes
These cookies spread while baking. I recommend leaving at least 1 1/2 -2 inches of space between each cookie. To be on the safe side, you might want to bake them in several batches while giving the dough ample room to spread out.Regular lemons may be substituted.