Add honey, water, and rosemary to a saucepan. Bring to a simmer over medium-low heat. Continue to simmer for 5 minutes. Remove from heat and cool to room temperature. Strain. Transfer the mixture to an airtight container and store in the refrigerator until ready to use. The syrup will stay fresh for up to 1 month.

To make the cocktail, add the Lillet, gin, simple syrup, and lemon juice to a shaker. Add a generous handful of ice. Shake vigorously for 20 seconds. Transfer the mixture to a rocks or highball glass filled with ice and top with club soda. Garnish with a sprig of rosemary and lemon slice.