Preheat the oven to 350F. Line two large baking sheets with parchment paper.
In a medium bowl, whisk together the flour, baking soda, baking powder and salt.
In the bowl of a stand mixer fitted with the paddle attachment, combine the sugar and lemon zest. Rub the mix between your fingers to release the oils. The mixture will be like wet sand and very fragrant.
With the mixer on low speed, add the melted butter. Beat until well incorporated.
Beat in the egg, egg yolk and vanilla extract until thoroughly combined.
Beat in the flour mixture just until combined. Scrape down the sides and bottom of the bowl, then beat again for 15 more seconds.
Using a large cookie scoop (I use #20, filling it about ¾ full) scoop the cookie dough, placing 6 cookie dough balls per baking sheet. You should have 12 cookies total.
Bake the cookies for 13-15 minutes, until the cookies are golden around the edges and the centers no longer look wet. Transfer to a wire rack to cool completely.
Lemon Frosting
In the bowl of a stand mixer fitted with the paddle attachment, combine the butter, cream cheese and lemon juice. Mix on medium speed for 2-3 minutes until smooth. Scrape down the sides and bottom of the bowl as needed.
On low speed, beat in the powdered sugar, lemon zest and salt.
Turn the speed to medium high and whip the frosting for 2-3 minutes, or until it's light, smooth and fluffy.