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1
Finely grate the ginger into a fine-mesh sieve set over a bowl.
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2
Press with the back of a spoon to squeeze out the juice (you'll need 1 tablespoon juice); set aside.
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3
Whisk the flour, cornstarch and salt in a medium bowl.
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4
Beat the butter and 1/2 cup confectioners' sugar in a large bowl with a mixer on medium-high speed until light and fluffy, 3 to 5 minutes.
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5
Beat in the lemon zest, 2 tablespoons lemon juice, the vanilla and ginger juice.
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6
Reduce the mixer speed to low; add the flour mixture and beat until just incorporated.
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7
Divide the dough between 2 sheets of plastic wrap; form into two 1-by-9-inch logs.
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8
Wrap and freeze until firm, about 1 hour.
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9
Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment.
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10
Unwrap the logs and slice into 1/4-inch-thick rounds; arrange 1 inch apart on the prepared baking sheets.
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11
Bake, switching the pans halfway through, until the edges are golden, about 13 minutes.
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12
Let cool5 minutes on the baking sheets, then transfer to racks to cool completely.
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13
Make the icing: Put the remaining 1/3 cup confectioners' sugar in a bowl; stir in 2 to 3 teaspoons lemon juice, a little at a time, until the icing is spreadable.
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14
Spread on the cookies and top with crystallized ginger; let set 30 minutes.
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15
Photograph by Andrew Purcell
