Finely grate the ginger into a fine-mesh sieve set over a bowl. Press with the back of a spoon to squeeze out the juice (you'll need 1 tablespoon juice); set aside. Whisk the flour, cornstarch and salt in a medium bowl.
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. Beat in the lemon zest, 2 tablespoons lemon juice, the vanilla and ginger juice. Reduce the mixer speed to low; add the flour mixture and beat until just incorporated.
Divide the dough between 2 sheets of plastic wrap; form into two 1-by-9-inch logs. Wrap and freeze until firm, about 1 hour.
Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment. Unwrap the logs and slice into 1/4-inch-thick rounds; arrange 1 inch apart on the prepared baking sheets. Bake, switching the pans halfway through, until the edges are golden, about 13 minutes. Let cool5 minutes on the baking sheets, then transfer to racks to cool completely.
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. Spread on the cookies and top with crystallized ginger; let set 30 minutes.