Beat the butter, sugar, egg yolk, and vanilla in the bowl of a stand mixer.
Add the flour, and mix on low speed until combined.
Cover the bowl with plastic wrap and refrigerate for at least 1 hour, or overnight.
Preheat oven to 350 F and line a baking sheet with parchment paper.
Once the dough is firm, shape it into 1" balls. Place the cookie dough balls on a cookie sheet, and use a small spoon to create a well in the center of each cookie. I use a 1/2 tsp measuring spoon for this step.
Bake the cookies for about 10-12 minutes, until the bottoms of the cookies begin to brown. You may need to take a spoon and deepen the wells in the cookies, since they may have flattened a bit (do this while they're still warm). Cool the cookies completely.
While the cookies cool, make your caramel by melting the butter and sugar in a small saucepan over medium heat. Once it reaches a boil, continue to cook the caramel for 1 minute, stirring constantly. If you have a candy thermometer, I let my caramel reach about 230 F.
Remove the caramel from heat and stir in the heavy whipping cream. Allow the caramel to cool to room temperature.
Drizzle the caramel into each cookie and sprinkle the tops generously with finishing salt.