Preheat oven to 350°F. Line one or more cookie sheets with parchment paper.
In a small bowl, whisk together 2 ¾ cups all-purpose flour, 1 teaspoon baking soda, ½ teaspoon baking powder, and ½ teaspoon salt; set aside.
In a large bowl, using a mixer on medium speed, beat 1 cup softened unsalted butter and 1 ½ cups granulated sugar until smooth and very fluffy, about 2–3 minutes.
Add 1 egg, ½ teaspoon pure vanilla extract, 3 tablespoons lime juice, and the zest of one large lime to the butter mixture; beat until combined.
Reduce mixer speed to low and gradually add the dry flour mixture, mixing just until incorporated and no streaks of flour remain.
Stir in ½ cup unsweetened toasted coconut until evenly distributed.
Scoop rounded teaspoonfuls of dough and roll each into a ball. Roll each ball in the ½ cup sugar for rolling until coated.
Arrange the coated dough balls on the prepared cookie sheets about 1 1/2 inches apart to allow for spreading.
Bake in the preheated oven for 8 to 10 minutes, or until the edges are set and the bottoms are lightly browned.
Let the cookies sit on the cookie sheet for 2 minutes, then transfer to a wire rack to cool completely.