Add the chopped peaches, sugar, and 2 cups of water to a medium saucepan. Bring to a boil over medium-high heat, stirring often so the sugar dissolves.

Reduce the heat and simmer for 5 to 10 minutes, or until the peaches are soft and beginning to break down. The syrup will deepen in color as it cooks.

Remove the pan from the heat and let the mixture cool to room temperature, about 30 minutes. Pour through a fine mesh strainer into a bowl or pitcher, pressing lightly on the solids to extract as much syrup as possible. You should have around 1 cup of peach syrup.

Juice the lemons and strain the juice to remove seeds and pulp. If you want to use lemon peel as a garnish, zest the lemons before juicing.

In a large pitcher, combine the peach syrup, lemon juice, and 3 cups of cold water. Stir well, then taste and adjust the balance of syrup, lemon juice, or water to your liking.

Serve over ice in tall glasses with lemon slices. Add ice to individual glasses rather than the pitcher if you expect leftovers, so the batch does not become watered down.