In the bowl of a stand mixer with paddle attachment or in a large bowl with a hand mixer, beat butter, sugar, and salt on medium-high speed until light and fluffy, 3-5 minutes. Mix in lime zest, lime juice, and orange cointreau.
With mixer running on low speed, add flour to butter mixture and beat to just blend, taking care not to over-mix.
Divide dough in half and place each half on large pieces of plastic wrap. Fold plastic wrap over dough to completely cover and use your hands to roll the dough into a smooth log, about 2″ in diameter. Repeat with second half of dough and then place in fridge to chill until firm, about 2 hours.
Preheat oven to 350 degrees and line a rimmed baking sheet with parchment paper. Taking one dough log out of the fridge at a time, brush the outside of each log with egg wash and roll in coarse sugar.
Slice each log into ½”-thick rounds and arrange on prepared baking sheet about 1″ apart. Bake for 11-13 minutes, until edges are just beginning to turn golden. Let cool slightly on pan before moving to wire racks to cool completely before icing.
Dip cooled cookies in tequila glaze coating about ½ of cookie. Sprinkle on sea salt and additional lime zest.