Let your butter come to room temperature. Wash two lemons. Use a citrus zester to remove the outer lemon skin, then juice them.
Cream butter and sugar in a large mixing bowl with an electric mixer at medium-high speed for about 2 minutes. Beat in eggs, lemon juice, and lemon zest till well combined.
Stir in flour, baking powder, and salt.
Roll dough into tablespoon sized balls and place on cookie sheets lined with a silicone mat or sprayed with cooking spray. Flatten the cookie dough balls with a glass dipped in sugar.
Bake at 350° for 9-10 minutes, or till barely golden brown on the bottom. Let sit 3-4 minutes on the pan, then remove to a cooling rack to cool completely.
Frosting:
Beat together butter, powdered sugar, salt, lemon zest, and lemon juice in a medium bowl till creamy, adding milk if needed. If you are piping the frosting, it needs to be thick. If you are just spreading it on, you can add more liquid.
Spread or pipe the frosting on top of the cooled cookies. I piped frosting on with a 1M star tip.
Notes
The cookies have a mild lemon flavor that is complimented by the lemon frosting. If you really love lemon, or aren't going to frost them, I'd double the amount of zest.
Cookies can be stored in an airtight container at room temperature for up to a week.