Adjust the oven racks to the upper-and lower-middle positions and heat the oven to 350 degrees F. Line two baking sheets with parchment paper; set aside.
In a large bowl, whisk together flour, baking powder, baking soda, and salt; set aside.
In the bowl of a stand mixer fitted with a paddle attachment, cream the butter and sugar at medium speed until light and fluffy, about 2 minutes. Scrape down the sides of the bowl with a rubber spatula as needed. Add the egg, lime zest, lime juice, milk and tequila; beat at medium speed until combined, about 30 seconds. Add the dry ingredients and beat at low speed until just combined, about 30 seconds, scraping down the bowl as needed.
Drop by rounded tablespoons of dough onto the prepared baking sheets, leaving about 3 inches between the cookies. Bake for 12 to 15 minutes, or until cookies are somewhat firm. Allow to cool on the sheets for 2 minutes, then transfer to wire cooling racks to cool completely before frosting.
To make the frosting: in the bowl of a stand mixer fitted with the paddle attachment, beat butter, 2 tablespoons lime juice, lime zest and powdered sugar until light and fluffy. Add more lime juice if needed to reach the desired consistency. After cookies are completely cool, ice them with a small spatula or butter knife.