Preheat your oven to 350°F (175°C) and line two baking sheets with parchment paper or silicone mats. Allow butter to sit at room temperature until soft enough to leave an indent when pressed.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt until evenly combined. Set aside.
Beat the softened butter and granulated sugar together with an electric mixer on medium speed for 2–3 minutes until light and fluffy, scraping down the sides halfway through.
Beat in the egg, then add the lime juice, lime zest, and vanilla extract, mixing on low speed just until incorporated. The mixture may look slightly curdled — that's normal.
Add the flour mixture in two or three additions, mixing on low speed after each just until no dry flour remains. Scrape the bowl bottom with a spatula; the dough will be soft and slightly sticky.
Cover the bowl with plastic wrap and refrigerate for 15–20 minutes to firm up the dough, making it easier to handle and helping cookies hold their shape during baking.
Place powdered sugar in a shallow bowl. Scoop dough into 1.5-tablespoon portions, roll into balls, then generously coat each ball in powdered sugar and place 2 inches apart on prepared baking sheets.
Bake for 11–13 minutes until the edges are set but the centers still look slightly underdone and characteristic cracks have formed in the sugar coating. Do not overbake.
Let cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely. Cookies will firm up but remain soft and chewy in the center.