Heat the oven to 350°F. Line the bottom and all 4 sides of a 9-inch square baking pan with 2 sheets of parchment paper, positioning them perpendicular to each and hanging over all 4 sides by at least 1 inch to form a sling.

Place 1 ½ cups granulated sugar and 2 teaspoons finely grated lemon zest in a stand mixer. Use your fingers to rub the zest into the sugar until fragrant. Add 2 sticks room temperature unsalted butter and ½ teaspoon kosher salt. Beat with the paddle attachment on medium speed until fluffy and lightened in color, 1 to 2 minutes.

Scrape down the sides of the bowl with a flexible spatula. Add ½ cup sour cream, 1 large egg, and 2 tablespoons lemon juice. Beat on medium speed until well-combined, about 1 minute (the mixture will look slightly curdled).

Scrape down the sides of the bowl again. Add 2 ½ cups all-purpose flour and beat on low speed until just fully incorporated. Transfer to the baking pan and spread into an even layer.

Bake until puffed, golden-brown around the edges, and light golden-brown on top, 25 to 35 minutes. Meanwhile, wash and dry the mixer bowl and paddle attachment.

Place the pan on a wire rack and let cool completely, about 2 hours. Once cooled, make the frosting.

Beat 6 ounces room temperature cream cheese, 3 tablespoons room temperature unsalted butter, 1 ½ teaspoons finely grated lemon zest, and ¼ teaspoon kosher salt together in the stand mixer with the paddle attachment on medium-high speed until light and fluffy, 2 to 3 minutes. Scrape down the sides of the bowl with a flexible spatula. Add 2 tablespoons lemon juice and beat until combined.

Add 4 ½ cups powdered sugar and beat on low speed until completely incorporated. Increase the speed to medium and beat until smooth and fluffy, 3 to 4 minutes.

Transfer the frosting onto the cookie slab and spread it into an even layer with an offset spatula. Garnish with more lemon zest if desired. Refrigerate until the frosting is set, about 20 minutes.

Grasping the excess parchment, lift the slab out of the pan and place on a cutting board. Trim the edges of the slab if desired, then cut into 16 pieces.