In a medium saucepan, bring 6 cups of water to a boil over high heat. Remove from the heat and add 1 cup of the peppermint leaves. Cover and let steep for 30 minutes to allow the mint to infuse the water. Strain the mint tea into a pitcher and refrigerate until cold
In a blender, puree the chopped peaches until smooth, adding a couple tablespoons of the mint tea, if necessary, to help puree the peaches. Strain the peach puree through a fine meshed sieve and chill the puree
Layer the crushed ice, thin slices of peach, and the remaining mint leaves in 4 tall glasses, filling the glasses about halfway.
Pour some peach puree into each glass, then stir in the chilled mint tea and serve immediately.