Preheat the oven to 350 degrees F. Line two baking sheets with parchment paper and set aside.

In the bowl of a stand mixer, combine the softened butter, granulated sugar, and powdered sugar. Cream together on medium speed until smooth and fluffy, about 2-3 minutes.

Add the oil, egg, lime juice, and coconut flavoring. Mix well on medium speed, scraping down the sides of the bowl as needed, until fully incorporated.

Add the flour, baking powder, and salt. Mix on low speed until thoroughly combined, scraping down the sides often. Do not overmix.

Using a large ice cream scoop (about 3 tablespoons), divide the dough into portions and place on the prepared baking sheets. Gently flatten each ball to about 1/2 inch thick with the palm of your hand.

Bake for 10-12 minutes, or until the centers are just set. Do not allow the edges to brown. The bottoms should be barely golden. Allow cookies to cool completely on the baking sheet before frosting.

In the bowl of a stand mixer, combine the softened butter and powdered sugar. Mix on low speed until combined.

Add the lime juice and lime zest. Increase to medium speed and whip until the frosting is smooth and fluffy, scraping down the sides often.

Once cookies are completely cool, spread a generous layer of lime frosting over each cookie. Sprinkle with additional lime zest and garnish with a small lime wedge to serve.