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To make the dough: In a large bowl or the bowl of a stand mixer fitted with the flat beater attachment, combine the sugars, lemon zest, and salt until the sugar smells fragrant and looks damp, about 30 seconds.
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Add the butter and beat on medium-low speed until smooth and well combined, 3 to 4 minutes. Scrape down the sides of the bowl, add the egg, vanilla, and lemon emulsion, and beat on medium speed to combine, about 1 minute. (It’s OK if the mixture looks curdled at first — it will smooth out.)
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Scrape down the sides of the bowl, add the flour, candied ginger, orange peel or orange zest, and ginger, and beat on low speed until combined, about 30 seconds.
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Transfer the dough to a sheet of parchment and divide it into two pieces (about 385g each). Place one half of the dough onto a second sheet of parchment. Shape each mound into a log that’s roughly 2" in diameter and 8" to 9" long. To make a more perfectly round circle, fold the parchment away from you to encase the log. Use your non-dominant hand to press a long, straight object (a ruler or a bench knife works well) in between the dough and the work surface, then use your dominant hand to gently tug the top piece of parchment away from you. This should pull the dough so that it butts up against the straight object and makes a very smooth, symmetrical log.
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Repeat with the second piece of dough, then place both logs in the refrigerator until very firm, 1 to 2 hours. (At this point, you can wrap the logs in plastic wrap and freeze for several weeks. Thaw in the refrigerator before slicing.)
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Preheat the oven to 400°F with racks in the upper and lower thirds.
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Working with one log of dough at a time, remove from the refrigerator, unwrap (reuse the parchment to line your baking sheet), and slice into 1/4"-thick rounds. Rotate the log about 90° (a quarter turn) every 3 to 4 slices to maintain a neat circular shape. Arrange on the baking sheets, placing the cookies about 2" apart.
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Bake the citrus shortbread for 11 to 13 minutes, rotating from top to bottom and front to back halfway through, until light brown on the outsides and golden in the centers. Let the shortbread cool on the baking sheets for about 10 minutes, then transfer to a wire rack to cool completely.
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Slice and bake your second log.
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To make the glaze: In a medium bowl, whisk together confectioners’ sugar and lemon juice. Slowly add more lemon juice by the teaspoon until you have a glaze that’s the consistency of molasses. Add lemon zest and salt, and stir to combine.
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Set the wire rack of citrus shortbread over the parchment-lined baking sheet. Dip each shortbread so that it's half covered in glaze, then top with candied ginger or orange peel while the glaze is still soft (you can dip a few shortbread, top them off, and repeat). Set the cookies aside for the glaze to firm up.
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Storage instructions: Store leftover citrus shortbread, well-wrapped, at room temperature for up to 7 days.
