In saucepan, bring sugar, water, pear and rosemary to a boil over medium heat, and continue boiling until sugar is dissolved. Reduce heat and simmer until pears are very tender and falling apart, 15 to 18 minutes.

Using an immersion blender or blender, whirl until smooth. Strain through a fine-mesh sieve into a bowl. Let cool. The Pear Rosemary Syrup makes ¾ cup. Store in the refrigerator for up to 1 week.

Arrange sliced pears inside a tall glass or wine glass; fill with ice. Top with pear syrup, club soda and rosemary; gently stir to combine.