Preheat oven to 325°F.
In the bowl of a stand mixer add the lime zest and sugar. Beat for about 2 minutes on high to release the lime oils into the sugar.
Add the butter, beat until well combined.
Add the eggs and vanilla, one at a time, beating well between additions.
Add the lime juice, beer, and olive oil, beating until well combined, scraping the bottom of the mixer to ensure all ingredients are well incorporated.
Stop the mixer and sprinkle with flour, baking powder, baking soda, and salt, stir until just combined.
Pour into an 8-inch loaf pan that has been greased.
Bake for 55 to 60 minutes or until cake is golden brown and toothpick inserted in the center comes back with just a few crumbs attached. Allow to cool completely.
Stir together the powdered sugar, mezcal, lime juice, and salt to make a thick paste. Spread over the top of the cake, chill until set about 3 hours. Cake is best made a day ahead of time.