Combine butter and sugars in the bowl of a stand mixer (or use a large bowl and an electric hand mixer) and beat until well-creamed.
With mixer on low, stir in egg yolk and vanilla extract, pausing to scrape down the sides and mixing until ingredients are well-combined.
In a separate bowl, whisk together flour, graham cracker crumbs, cornstarch, and salt.
Gradually add the flour mixture to your wet ingredients, being sure to scrape the sides and bottom of the bowl so that all ingredients are well-combined.
Scoop cookie dough into 2-teaspoon-sized balls and roll until smooth (so that the dough is round and there are no cracks/seams in the cookie dough — this will help keep your thumbprints from cracking).
Use your thumb or the rounded back of a teaspoon to gently press an indent in the center of the cookie dough. Repeat until all of the dough has been used.
Chill imprinted dough in freezer for 30 minutes.
Preheat oven to 350F (175C) and line cookie sheets with parchment paper (if using. An ungreased, non-stick cookie sheet will also work).
Place chilled cookie dough on sheet spacing at least 1″ apart.
Bake on 350F (175C) for 10-11 minutes and allow to cool completely on pan. Wait for cookies to cool completely before preparing filling.
Key Lime FillingIn a small saucepan over medium-low heat, combine white chocolate chips, condensed milk, and key lime juice.
Stir frequently until chocolate is melted and ingredients are well-combined. Stir in lime zest and stir until mixture just begins to bubble.
Reduce heat to low and spoon filling into thumbprints of cookies (I use a 1-tsp sized cookie scoop, it makes the process much easier and less messy!).
Allow to cool completely before serving (filling may still be a little tacky to touch, even when cooled completely. This is normal).