Preheat the oven to 350F. Line a baking sheet with parchment paper.
In the bowl of a stand mixer fitted with a paddle attachment, mix the butter and sugar on medium speed for 2 minutes until light and fluffy.
Add the egg, vanilla, key lime juice and lime zest and continue mixing for 1 more minute, scraping the sides of the bowl as necessary. Mix in the salt and baking soda until incorporated.
Turn the speed to low and add the flour and graham cracker crumbs until the dough comes together.
Roll the dough into 1-inch balls and place 2 inches apart on the baking sheet. Bake for 9 to 10 minutes, until the edges are lightly golden.
Transfer the cookies to a wire rack to cool.
Place the white chocolate in a microwave-safe bowl. Heat on full power for 25 seconds. Remove the bowl from the microwave and stir the chocolate. Repeat this process until the white chocolate is melted and smooth. Transfer the melted chocolate to a small ziptop bag. Snip one corner of the bag off with scissors and drizzle the white chocolate on top of the cookies. Allow the white chocolate to set completely.
Store the cookies airtight at room temperature for up to 3 days.