Step 1

Process ginger in food processor until finely chopped, about 30 seconds, scraping down sides of bowl as needed.

Step 2

Heat sugar and water in small saucepan over medium heat, whisking often, until sugar has dissolved, about 5 minutes; do not boil. Off heat, stir in chopped ginger and ground ginger and let cool to room temperature, about 30 minutes. Cover and refrigerate for at least 12 hours or up to 24 hours.

Step 3

Set fine-mesh strainer over medium bowl and line with triple layer of cheesecloth that overhangs edges. Transfer ginger mixture to prepared strainer and let drain until liquid no longer runs freely, about 10 minutes. Pull edges of cheesecloth together to form pouch, then firmly squeeze pouch to extract as much syrup from pulp as possible; discard pulp. Stir in lemon juice. (Ginger Syrup can be refrigerated in airtight container for up to 1 month. Shake gently before using.)