Preheat oven to 375 degrees F. Grease a 9x13 inch metal baking pan and line with parchment paper for easy removal.
To make the lemon cookie bars, beat butter, sugar, brown sugar, and lemon zest together until fluffy and well combined in a stand mixer with the paddle attachment.
1 cup salted butter, 1 ¼ cups granulated sugar, 1/4 cup light brown sugar, Zest from 1 lemon
Add egg and egg yolks. Beat until eggs are well incorporated, about 2 to 3 minutes.
Add flour, baking powder, and salt. Mix in until just combined.
3 cups sifted all purpose flour, 1 teaspoon baking powder, 1/2 teaspoon kosher salt
Add cookie batter to the prepared pan in an even layer.
Bake for 15-20 minutes, or until the edges are lightly golden brown.
Let the bars cool in the pan until they reach room temperature.
To prepare the frosting, beat butter until light and fluffy.
1 cup salted butter
Add lemon zest, powdered sugar, and salt. Mix until combined.
Zest from 1 lemon, 3 cups powdered sugar, Pinch of kosher salt
Stir in vanilla extract and lemon juice.
1 teaspoon vanilla extract, 2 teaspoons freshly squeezed lemon juice
Add milk (or heavy cream) until the frosting is spreadable and reaches your desired consistency.
2-3 tablespoons milk or heavy cream
Spread frosting on top of the cooled cookie bars.
Decorate with additional lemon zest and lemon slices.
Notes
Butter: If using unsalted butter, increase the salt in the bars by 1/2 teaspoon and the frosting by 1/4 teaspoon.
Lemon juice: It's important to use fresh lemon juice as bottled lemon juice can have an off taste to it.
Flour: Weighing the flour is the most accurate method of measurement (click "metric" just above the ingredients for the weights). If not weighing the flour, sift it, lightly spoon into the measuring cup, and level. Avoid packing the flour as that will give you dry, crumbly bars.
Kosher salt: Sea salt works well as a substitute. If you don't have kosher salt or sea salt, use half the amount of table salt.
Milk: Using heavy cream will give you the lightest, fluffiest frosting.