To make the Lime-Mint Syrup, bring the sugar and water to a boil cook until completely dissolved, only about 1 minute. Remove from heat, stir in the lime zest and mint, and let sit for about 30 minutes. Strain, cover and refrigerate until chilled.
Muddle mint and lime with the crushed ice in the bottom of a large pitcher. Add chilled Lime-Mint Syrup, rum, lime juice, and more ice. Top off with chilled club soda, to taste.
Pour into glasses over crushed ice and serve. Garnish each cocktail glass with a lime slice and mint sprigs.