In a large bowl, whisk together 1 cup (200 grams) granulated sugar, mint, and lime zest until well combined and fragrant. Whisk in melted butter, egg, and egg yolk. Whisk in lime juice.
In a medium bowl, whisk together flour, salt, and baking soda. Add flour mixture to sugar mixture in two additions, folding until well combined after each addition. Fold in green gel food coloring (if using) until desired color is reached. Cover and refrigerate for at least 2 hours.
Preheat oven to 325°F (170°C). Line 2 rimmed baking sheets with parchment paper.
In a small bowl, place remaining granulated sugar. In another small bowl, place confectioners’ sugar.
Divide dough into 36 portions (about 18 grams each). Shape into balls, and roll in granulated sugar; roll in confectioners’ sugar. Place 1½ to 2 inches apart on prepared pans.
Bake until edges and tops are just set and cracks have formed, 8 to 10 minutes. Let cool completely on pans.
Spoon Crème Fraîche Frosting into a large pastry bag fitted with a 3/8-inch round piping tip (Ateco #804). Pipe frosting onto flat side of half of cookies. Place remaining cookies, flat side down, on top of frosting. Serve immediately.