Getting Ready: Adjust rack to lower-middle position and preheat oven to 350°F (180°C). Line a 9- by 13-inch anodized aluminum baking pan with parchment paper, covering bottom and long sides of pan. (It’s all right to leave the short sides exposed.) Lightly grease pan with nonstick cooking spray.
For the Syrup: In a 2-quart nonreactive saucepan over medium-high heat, simmer together sugar, apple juice, ginger, and lemon peel until syrup reaches 234°F (112°C) on an instant-read thermometer. Remove syrup from heat and add lemon juice and Applejack. Set aside and allow to cool to room temperature.
For the Cake: In the bowl of a stand mixer, combine eggs, sugar, and salt. Using a whisk attachment, whip on high until eggs become thick, pale, and ribbon-y, about 10 minutes. With mixer running on medium, slowly drizzle in oil in a thin stream until fully incorporated and thick. Whisk in grated ginger, about 10 seconds, then shut off mixer and remove bowl.
