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Preheat oven to 350°F.
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Line baking sheets with parchment.
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Cream butter and 1/3 cup sugar until fluffy.
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Add the lime juice, zest and vanilla and beat until fluffy.
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In another bowl, combine together flour, cornstarch and salt.
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Add to butter mixture and beat until combined.
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Roll into a log and chill for 1 hour.
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Slice logs into 1/8-inch rounds.
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Place rounds on sheets and bake until golden, 11-15 minutes.
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While still warm, place cookies in bag with the rest of the powdered sugar and toss gently to coat.
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They are tender and may break apart.