In a medium saucepan, combine the diced peaches, sugar, and ground cardamom. Cook over medium heat, stirring occasionally, until the peaches have softened and released their juices, about 5-7 minutes.

Remove the saucepan from heat and let the mixture cool for about 10 minutes. Once cooled, strain the mixture through a fine mesh sieve into a bowl, pressing down on the solids to extract as much liquid as possible. Discard the solids.

In a large pitcher, combine the peach-cardamom syrup with the freshly squeezed lemon juice. Stir well to combine.

Add the cold water to the pitcher and mix until well combined. Taste and adjust the sweetness by adding more sugar if desired.

Chill the lemonade in the refrigerator for at least 30 minutes before serving.

Serve the lemonade over ice, garnished with lemon and peach slices.