Preheat your oven to 350°F (175°C) and line two baking sheets with parchment paper or silicone baking mats, placing the oven rack in the center position.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt until evenly combined, then set aside.
In a large bowl, beat the softened butter and granulated sugar with an electric mixer on medium speed for 2–3 minutes until light and fluffy, scraping down the sides as needed.
Beat in the egg until fully incorporated, then mix in the sour cream, key lime juice, key lime zest, and vanilla extract on low speed until just combined.
With the mixer on low, gradually add the dry ingredients to the wet mixture and mix until just combined with no flour streaks remaining — do not overmix.
Using a 2-tablespoon cookie scoop, drop rounded portions of dough onto the prepared baking sheets, spacing them about 2 inches apart for a total of about 24 cookies.
Bake for 10–12 minutes until the edges are just set and the tops look slightly dry but not browned — the cookies will look slightly underdone, which is perfect.
Let the cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack and cool completely before glazing.
Whisk together the powdered sugar and 2 tablespoons of key lime juice, adding more juice a teaspoon at a time until the glaze is thick but pourable.
Dip or drizzle the glaze over each cooled cookie, then optionally sprinkle with key lime zest or graham cracker crumbs, and allow the glaze to set for 15–20 minutes before serving.