Combine the sliced peaches, sugar, 1 cup of water, and ginger slices in a small saucepan over medium heat. Stir to help the sugar start dissolving.

Bring the mixture to a gentle simmer, then reduce the heat to medium-low. Cook for 12 to 15 minutes, stirring occasionally, until the peaches are very soft and completely broken down and the liquid looks syrupy and slightly thickened.

Remove from the heat and let the mixture cool for 10 minutes so it is easier to handle. Pour it through a fine mesh strainer set over a large bowl or pitcher. Press the peach and ginger solids firmly with the back of a spoon to squeeze out every bit of flavor, then discard the solids.

Let the syrup cool completely to room temperature, about 20 minutes, or speed it up by setting the bowl in a larger bowl of ice water for 10 minutes.

Once the syrup is cool, add the fresh lemon juice and 4 cups of cold water to the pitcher with the syrup. Stir well to combine. Taste and adjust: add a splash more lemon juice for more tartness, or a little more water if it is too sweet or strong.

Pour over ice in tall glasses and garnish with a lemon slice if you like. Serve cold, or gently warm a mug of it on the stovetop in fall and winter without adding ice.