Combine the sugar, water, and rosemary sprigs in a small saucepan and simmer over low heat for 5 minutes, stirring until the sugar dissolves.
Remove from heat, let the rosemary steep for 10 minutes, then strain out the sprigs and let the syrup cool.
If using fresh peaches, blend them until smooth.
Add the peach puree, 2 tablespoons of rosemary syrup, and lemon juice to a shaker with ice. Shake for 15 seconds until well chilled.
Strain into a glass filled with fresh ice.
Top with club soda, pouring slowly to preserve the fizz.
Garnish with a rosemary sprig and a peach slice, and serve immediately.