In a medium saucepan, combine peeled and chopped peaches, sugar, and water over medium-high heat. Stir frequently as the mixture comes to a boil, allowing the sugar to dissolve completely and the peaches to start releasing their juices, which should take about 5 minutes.

Once boiling, reduce the heat and let the mixture simmer gently for 5-10 minutes, until the peaches are soft and mushy. You'll know it's ready when the mixture thickens slightly and the peaches have completely broken down. Remove from heat and let it cool for about 30 minutes.

After cooling, pour the mixture through a fine sieve into a bowl or pitcher, using a spatula to press down on the pulp. Discard the solids and set the freshly made peach syrup aside for later use.

While the syrup cools, juice two fresh lemons and consider zesting them for a garnish and extra flavor. You should have about a third of a cup of lemon juice.

In a large pitcher, combine the peach syrup, freshly squeezed lemon juice, and water, mixing well to ensure an even distribution of flavors. Adjust the amount of water based on your desired strength.

Give the mixture a taste to determine if it needs more sweetness or tartness. If you prefer a sweeter beverage, add a bit more peach syrup; for extra tang, incorporate a splash more lemon juice. Stir thoroughly, then refrigerate for about 30 minutes to chill.

When ready to serve, fill glasses with ice cubes, then pour the chilled peach lemonade over the ice. Garnish with lemon slices or mint leaves.