In the bowl of a stand mixture using the paddle attachment, or in a large bowl using a hand mixer, beat the butter, one-half cup plus 1 tablespoon sugar, eggs and vanilla at medium speed until well incorporated, about 2 minutes (it will not be fluffy).
Meanwhile, grind the coconut in a food processor until finely chopped, about 10 seconds (you should have a scant 1 2/3 cups).
In a medium bowl sift together the flour, salt and baking soda. Mix the combined dry ingredients, one-half at a time, into the butter mixture until thoroughly combined, scraping down the sides of the bowl as needed. Add the coconut, mixing well.
Divide the dough in half, wrap each half in plastic wrap, and flatten into a disc (the dough will be very sticky). Chill for 2 hours or longer to firm the dough.
Heat the oven to 375 degrees. Place the remaining two-thirds cup of sugar in a small bowl. Using one round of dough at a time (keep the rest chilled), shape the dough into 1-inch balls and roll in the sugar. Place the dough 2 inches apart on a lightly buttered baking sheet. Lightly grease the bottom of a pint glass, or similar glass, with a little butter, then dip the glass into the sugar to coat. Press the cookie with the bottom of the glass until it is about one-eighth inch thick and 2 1/2 inches wide. Repeat with all the balls, and make sure the glass is well-sugared each time before pressing, as the dough will be sticky. Repeat with the second half of the refrigerated dough, forming and pressing the balls.
Bake the cookies, one pan at a time and in the center of the oven, until they are lightly golden on the edges but still white in the middle, about 10 minutes. Rotate the pan halfway through for even baking. Remove the cookies immediately to a wire rack to prevent them from sticking to the baking sheet. Cool completely.
To assemble, spread one-third cup ice cream (slightly softened) over the bottom of one cookie, and carefully top with another cookie. Press firmly but gently to avoid breaking the cookies. Run a knife or offset spatula around the edges to clean up any ice cream hanging over the edge. Freeze for at least 1 hour to firm the ice cream. Repeat with all the cookies.