Beat the butter in the bowl of a stand mixer on high for 1 minute. Add the sugar and continue beating for 3 minutes more. Beat in the lime juice, zest, and vanilla until it is all incorporated. Add the flour and nuts. Scrape the bottom of the bowl to make sure that all of the butter is mixed in.
Working with one-half of the dough at a time, roll out on a lightly floured surface to ¼” thickness. Be gentle with the dough. Roll it, then lift it and turn it and roll it again.
Cut cookies out of the dough and place 1” apart on an ungreased cookie sheet. Bake for 12-15 minutes until the cookies are just beginning to brown on the edges.
Remove the baked cookies from the cookie tray and cool completely on a wire rack.
Make the glaze by putting all of the ingredients in the bowl of a mixer and beating on high until it is smooth. This may take as long as 3 minutes.
Glaze the cooled cookies and decorate, if desired, with sprinkles, lime zest or chopped hazelnuts.