Prepare the peaches: Wash the peaches thoroughly. Bring a medium pot of water to a boil. Gently place the peaches in the boiling water for 30-60 seconds (blanching) to loosen the skins. Immediately transfer them to an ice bath. Peel the skins off, remove the pits, and coarsely chop the peach flesh.

Simmer the mixture: In a medium saucepan, combine the chopped peaches, 3 cups of water, granulated sugar, and optional cinnamon. Bring the mixture to a boil over medium-high heat, then reduce the heat to low and simmer, uncovered, for 10 minutes, stirring occasionally until the peaches are very soft.

Blend the ingredients: Carefully transfer the hot peach mixture to a high-powered blender. Add the fresh lemon juice. Blend until completely smooth. Scrape down the sides as necessary to ensure no lumps remain.

Strain the juice: Place a fine-mesh sieve over a large bowl or pitcher. Pour the blended peach mixture through the sieve. Use the back of a spoon or a ladle to press down on the solids to extract as much liquid as possible. Discard the remaining pulp.

Chill and serve: Taste the strained juice and adjust sweetness if necessary (add a tablespoon more sugar if needed, dissolving it in a small amount of hot water first). Let the juice cool to room temperature, then cover and chill in the refrigerator for at least 2 hours before serving over ice.