Christmas Caipirinhas

• Peel from 4 large limes (green part only; removed with vegetable peeler in 2-inch-long strips)
• 1 cup fresh lime juice
• 8 1/4-inch-thick slices peeled fresh ginger
• 1/4 cup sugar
• 30 large fresh mint leaves plus 10 fresh mint sprigs for garnish
• 1 cup Brazilian rum (such as cachaça)
• 3 cups ice cubes plus additional for serving
• 5 cups ginger beer or ginger ale