Place half of the mint sprigs, the tea bags, and the peach slices in a heat-resistant half-gallon pitcher.
In a large saucepan over medium-high heat, bring the water to a boil, then pour it into the pitcher. Let steep for 10 minutes.
Remove the mint and tea bags, discard. Let the tea cool completely, then cover and refrigerate until ready to serve. Strain the iced tea through a fine mesh sieve.
To serve, pour the tea into ice-filled glasses, add a few peach slices, sprinkle over a little bit of the remaining mint leaves, and stir in syrup to taste.