Preheat the oven to 350 degrees F, and line several baking sheets with parchment paper. Using an electric mixer cream the butter, sugar, and vanilla together until light and fluffy, 3-5 minutes. Scrape the bowl and beat again for 30 seconds.
Sift the flour and salt together into a separate bowl. Then turn the mixer on low, and add the flour mixture a little at a time, until the dough comes together. Dump the dough onto a large sheet on plastic wrap. Flatten into a disk and wrap well. Place the dough in the fridge to chill for at least 30 minutes.
Roll the dough out on a very well floured surface. Roll until it is 1/3-inch thick. Using a floured cookie cutter, cut out cookies and place on the parchment paper lined cookie sheets two inches apart. Try to cut out as many as possible from the first sheet of dough. Each time you gather and re-roll the dough, the cookies will get a little tougher.
Bake for 20 minutes, until barely golden. Remove from the oven and immediately sprinkle with sea salt. Allow the cookies to cool completely.
For the Chocolate:
Set a glass bowl above a small sauce pot with about one inch of water to create a double-boiler. Heat the pot over medium heat. Place broken chocolate pieces and 1/2 tablespoon butter in the bowl. Stir as the chocolate melts.
Remove the bowl from the pan. Dip each cookie half way in the chocolate. Use a spatula to scrape the excess chocolate off the back. Place the cookies back on the parchment paper or on a wire rack. Allow them to cool and harden. If you need the chocolate to set faster, place in the fridge for 15 minutes. Store in an air-tight container.