In a large bowl, combine the lemon zest and sugar by rubbing both ingredients together. Doing so will release the natural lemon oils from the zest creating a robust lemon flavor.
To the same bowl, using a hand whisk or stand mixer, cream the sugar and butter together until pale and fluffy, approximately 3-4 minutes.
Pour in the lemon juice, whole egg, and egg yolk, then whisk again until fully combined.
To the wet ingredients, gently pour in the flour and baking soda. Switch to a spatula to fold in and combine. STOP mixing when the flour is fully incorporated.
Line a baking sheet with parchment paper, then using a large cookie scoop, scoop the cookie dough batter into balls, then place on the baking sheet.
Repeat until all the batter is formed into cookie balls, then chill for an hour in the fridge.
About five minutes before the chill time is up, preheat the oven to 350º, remove the cookies from the fridge, and bake the cookies in batches of six, evenly spaced. Bake for 11 minutes, then let cool and transfer to a wire rack to finish cooling for 30 minutes.
If you are adding the glaze, whisk the powdered/confectioner's sugar with the half and half and the lemon juice until well combined. Using a spoon, drizzle the glaze on top of the cooled cookies, let set for 10 minutes before serving.
Notes
Do not over-mix the batter. If you over-mix it will work the gluten too much in the flour, causing the dough to become dense and tough. We want chewy and soft. My trusty tip is to switch to a sturdy rubber spatula to fold in the dry ingredients. Works like a charm!Use Room Temperature Ingredients - The softened and room temperature butter and eggs gives these cookies their chewy texture and rich flavor. Let the cookies cool all the way before adding the lemon glaze.