For the cookies: Preheat the oven to 350 degrees F.
Using a hand or stand mixer, cream the butter and granulated sugar until combined. Add the egg yolks and mix until combined. Add the lemon and lime zest. Add the flour and mix until just combined. Next, add the citrus juice (your choice) and coconut extract and mix the dough until it comes together.
Scoop out tablespoons of dough, roll into balls between your hands and place on a baking sheet. Bake until not quite starting to turn brown, 12 to 13 minutes. Remove from the oven but keep the cookies on the baking sheet for 3 to 4 minutes. Use a spatula to move them to a cooling rack to cool completely.
For the icing: Add the powdered sugar, cream of coconut, lemon and lime zest, milk, coconut extract, salt, egg white and lemon and lime juice to a large bowl. Whisk thoroughly until combined, adding either more powdered sugar or more juice until it reaches a pourable but still thick consistency.
Drizzle the icing across the cookies in several lines, and then repeat in the other direction. Sprinkle the cookies with extra zest before the icing sets.