How to Make It
Stir together sugar, water, and mint leaves in a small saucepan; bring to a simmer over medium-high, stirring to dissolve sugar. Simmer, undisturbed, until mint wilts and syrup tastes minty, about 3 minutes. Remove and discard mint. Chill syrup until cold, about 30 minutes.
Stir together mint syrup, rum, and lime juice in a large pitcher. Chill at least 1 hour or up to 3 days.
When ready to serve, gently stir club soda into mixture in pitcher, and add lime rounds. Pour evenly into 6 ice-filled glasses. Garnish glasses with lime peel strips and mint sprigs.