Preheat the oven to 350°F. Line a nine inch square baking pan with foil, leaving at 2 inch overhang on two sides.

Beat the butter, vanilla and salt in a large bowl with the mixer on medium speed until smooth and creamy, about 2 minutes. Reduce the mixer speed to low. Add confectioners sugar and beat until just incorporated. Add the flour in two batches and beat until smooth. Stir in half a cup of chocolate chips with a wooden spoon.

Lightly dust your hands with flour and press the dough into the prepared pan. Score the top with a knife to make nine strips, then score the strips in half to make 18 pieces. Bake until the edges are firm and the top is dry, 40 to 45 minutes. Immediately rescore the cookies, then let cool 20 minutes in the pan. Lift the foil to remove the shortbread from the pan and transfer to a rack to cool completely.

Cut the shortbread into pieces along scored lines. Put the remaining half cup chocolate chips in a small microwave safe bowl. Microwave in 30 second intervals, stirring, until melted. Drizzle over the shortbread and let stand until set, about 1 hour.