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.

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.

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.

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.

Repeat with the second piece of dough, then place both logs in the refrigerator until very firm, 1 to 2 hours.

Preheat the oven to 400°F with racks in the upper and lower thirds.

Working with one log of dough at a time, remove from the refrigerator, unwrap, and slice into ¼"-thick rounds.

Arrange on the baking sheets, placing the cookies about 2" apart.

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.

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.

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.

Set the cookies aside for the glaze to firm up.