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Pour 6 cups of water into a medium saucepan. Cover and bring to a boil over high heat. Remove from heat and add 1 cup of mint leaves.
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Cover and steep for 30 minutes to allow mint to infuse water.
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Strain mint tea into a pitcher, discarding leaves, and refrigerate until cold.
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Meanwhile, chop 2 peaches coarsely and place them in a blender. Puree until smooth, adding a couple tablespoons of mint tea to help create the puree. Strain peach puree through a fine-mesh sieve and chill puree. You should have about 1 cup of puree. It will have a peach color when first blended, and will turn into a brownish tea color once chilled.
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Thinly slice remaining peach. In 4 tall glasses, layer crushed ice, thin slices of peach, and remaining 1/4 cup mint leaves, filling glasses about halfway.
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Divide chilled mint tea among 4 glasses. Then stir in peach puree and serve immediately.