In a small saucepan, combine the sugar, water, and 8 mint sprigs and bring to a boil. Simmer the mixture over moderate heat until the sugar dissolves.Remove from the heat and let cool slightly, about 10 minutes.
Strain the mint syrup into a blender and puree with the 1 cup of mint leaves.
In a large saucepan, bring the cold water to a boil. Remove from the heat, add the tea bags and let steep for 5 minutes. Remove the tea bags and let cool to room temperature. Transfer to a bowl and refrigerate until chilled.
Pour the tea into a pitcher, add the lemon juice and stir in 3/4 cup of the mint syrup, or more to taste; refrigerate the remaining syrup for later use. Serve in 8 tall glasses over ice, garnished with a mint sprig and a slice of lemon.
MAKE AHEAD The iced tea can be refrigerated in a covered pitcher for up to 2 days. NOTES To make the tea faster, steep the tea bags in 4 cups of boiling water for 10 minutes; then stir in 4 cups of ice water. Proceed with the recipe.