Preheat your oven to 350°F. Line a baking sheet with parchment paper.
In a medium bowl, whisk together the flour, baking soda, and salt. Set aside.
In a large bowl, cream together the softened butter, granulated sugar, and brown sugar until light and fluffy, about 2-3 minutes.
Add the egg, vanilla extract, key lime juice, and lime zest to the butter mixture. Mix until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
In a separate bowl, mix the sweetened condensed milk, powdered sugar, key lime juice, and lime zest until smooth. This will be your filling.
Scoop out about 2 tablespoons of cookie dough and flatten it in your hand. Place about 1 teaspoon of the filling in the center, then fold the dough over to seal it. Roll it into a ball and place it on the prepared baking sheet. Repeat with the remaining dough and filling.
Bake for 10-12 minutes, or until the edges are lightly golden. Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.