Make lemon ginger syrup. Use a vegetable peeler to remove thin strips of lemon peel. Set aside. Cut the lemon in half and squeeze out the juice (you’ll need 2 small lemons to get the right amount). Add the lemon juice, water, and sugar to a small saucepan over medium-high heat, stirring constantly until the sugar dissolves.
Simmer. Reduce to medium-low heat, then add the fresh ginger pieces and lemon peel. Simmer for ten minutes while stirring occasionally, then remove from the heat.
Strain syrup. Run the syrup through a fine mesh strainer to remove the peels and sliced ginger. Let cool completely before putting the lid on and storing.
Blend. Add the green apple slices, water, and a squeeze of lemon juice to a high-speed blender. Blend until smooth, then run through a fine mesh strainer, pushing against the pulp with the back of a spoon to get as much juice out as possible.
Assemble drink. Mix the tart apple juice and lemon ginger simple syrup in the bottom of a lowball glass until well combined. Add ice and fill the glass the rest of the way with sparkling water. Garnish with a lemon peel or piece of sliced ginger from the syrup.