Heat oven to 350 degrees. Line two baking sheets with parchment; set aside. In an electric mixer, use paddle to mix butter and sugar on medium-high speed until light and fluffy, about 5 minutes, scraping down sides of bowl twice. Add egg; mix on high speed to combine. Add zest; mix to combine.
In a bowl, whisk together flour, ground ginger, baking soda, salt, and crystallized ginger; add to butter mixture; mix on medium-low speed to combine, about 20 seconds.
Using two spoons, drop about 2 teaspoons of batter on baking sheet; repeat, spacing them 3 inches apart. Bake for 7 minutes. Sprinkle cookies with sugar, rotate sheets between oven shelves, and bake until just golden, about 7 minutes more. Slide parchment with cookies onto a wire rack; let cool 15 minutes. Store in an airtight container.