In the bowl of a stand mixer fitted with a paddle attachment, beat the butter, flour, sugar, vanilla and kosher salt until the crumbs form and the mixture just comes together, about 1 minute. The dough will be very crumbly.
Turn out onto a workspace lined with a piece of plastic wrap. Gather dough into a 3½” x 1⅕” rectangle and wrap, pressing to hold together. Cover in the plastic wrap and refrigerate until firm, about 1 hour.
Cut the shortbread into ½” slices and arrange on a parchment paper-lined baking sheet about 1” apart.
Use a fork and poke holes in the shortbread in an even pattern. Bake the shortbread until lightly golden brown on top, about 20 minutes. Let cool.
Melt chocolate using the double boiler method. Bring a medium saucepan of water to a simmer over medium-high heat. Set a heatproof bowl over the top and add the chocolate. Stir constantly with a rubber spatula until the chocolate melts.
Dip half the shortbread into the chocolate and set on a parchment paper-lined baking sheet. Sprinkle with flakey sea salt and allow to dry completely, about.1 hour.