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Preheat the oven to 350 degrees F. Line a 9-inch-square baking pan with foil, leaving a 2-inch overhang on 2 sides.
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Beat the butter, vanilla and salt in a large bowl with a mixer on medium speed until smooth and creamy, about 2 minutes. Reduce the mixer speed to low; add the confectioners' sugar and beat until just incorporated. Add the flour in 2 batches and beat until smooth. Stir in 1/2 cup chocolate chips with a wooden spoon.
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Lightly dust your hands with flour and press the dough into the prepared pan. Score the top with a knife to make 9 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.
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Immediately re-score 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.
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Cut the shortbread into pieces along the scored lines. Put the remaining 1/2 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.