Preheat the oven to 375°F.
To make the cookies:
In a bowl of a mixer, cream together the butter and sugar. Beat in the eggs.
In a medium-sized bowl, stir together flour, cream of tartar, baking soda, and salt. Gradually add the dry ingredients to the mixer. Mix on medium speed just until all ingredients are combined. Add the lime zest and the toasted coconut. Mix just until incorporated.
Prepare the sugar dusting for the cookies by stirring together sugar, lime juice, and lime zest in a shallow dish.
Roll the cookie dough into 20 large balls about 2 inches in diameter. Roll the cookie dough balls in the sugar dusting to coat each.
Place the cookies on a baking sheet leaving space around them to spread, about 6 or 9 cookies to each sheet pan.
Bake for 12 minutes, or until the cookies begin to brown around the edges and appear firm in the middle. Remove from the oven and let cool on the pan for 2 to 3 minutes, transfer to a cooling rack and let them cool completely.
To make the sandwiches:
Pour the toasted coconut in a shallow pan or on a plate.
Place a cooled cookie, bottom side up, on a flat surface. Top with about ½ cup of vanilla ice cream. Use a spoon to gently press and spread the ice cream over the cookie. Top with another cookie, bottom side down. Press gently so that the ice spreads to the edge. Roll the edges of the ice cream sandwich in the coconut. Repeat the steps with each ice cream sandwich.
Wrap in plastic wrap and freeze until the ice cream is firm again, about 3 hours.