Add the rye and leaves from one sprig of rosemary to a cocktail shaker and muddle.
Add remaining ingredients and ice. Shake for 30 seconds and double-strain into a rocks gass with one large ice cube.
Garnish with one sprig of rosemary and a twist of lemon peel.
Combine sugar, water, and grated ginger in a small saucepan. Bring to a simmer over medium heat, stirring often until sugar is dissolved. Simmer for 10 minutes, reducing heat as needed, to allow ginger to infuse. Remove from heat and let cool to room temperature before using. Syrup will keep in an airtight container in the fridge for up to two weeks.