Prepare the limes by grating 3 tsp of lime zest in one bowl, and the juice (about 4 Tbsp) of lime juice in another bowl.
Beat the soft butter in a large bowl; add the sugar and mix well. Add in the zest and 3 Tbsp of lime juice, and vanilla. (Save the rest of the lime juice for the frosting).
Beat in the flour and nuts; divide dough in half.
On a lightly floured surface, roll half of the dough to 1/4-inch thick. Using 2-inch square or round cookie cutters (or your favorite cookie cutter), cut out the dough and place 1-inch apart on ungreased cookie sheet. Repeat the process for the 2nd half of dough and place on cookie sheet.
Bake until the edges are lightly brown, 8-10 minutes. Remove from the oven and allow to cool on wire racks.
Make the frosting:
Whisk together the lime glaze ingredients, adjusting liquid or powdered sugar to desired thickness. Allow the glaze to sit for several minutes and then spread the glaze on the cookies.
It’s okay if the icing runs over a little bit. Sprinkle it with remaining lime zest. Allow the cookies to sit for 1 hour before eating (to allow the frosting to set).
Store remaining cookies in an airtight container for 4-5 days.