Combine chopped peaches, granulated sugar, and water in a small saucepan over medium heat. Bring to a simmer, stirring occasionally, and cook for 6–8 minutes until peaches are very soft and sugar is dissolved.

Macerate and mash

Remove from heat and add the 1 tablespoon fresh lemon juice. Use a potato masher or the back of a spoon to mash the peaches in the saucepan to release more juice and flavor (or pulse once or twice in a blender for a smoother syrup).

Strain and cool

Pour the mixture through a fine-mesh sieve into a bowl or measuring cup, pressing on solids with a spoon to extract as much liquid as possible. Discard or reserve the solids for another use. Let the strained syrup cool to room temperature, then chill in the refrigerator for at least 30 minutes before assembling the drink. Yield: about 3/4–1 cup syrup.

Cooler Assembly

Fill glasses

Divide the ice cubes between two tall glasses (about 1 cup ice per glass total).

Measure syrup and lemon

Add 3/8 cup (about half of 0.75 cup total) of the chilled peach simple syrup to each glass. Add 1/2 teaspoon fresh lemon juice to each glass to brighten the flavor.

Top with sparkling water

Pour 3/4 cup cold sparkling water or club soda into each glass over the syrup and ice. Gently stir once or twice to combine, taking care not to flatten the bubbles.

Garnish and serve

Tuck three fresh mint leaves into each glass (lightly clap between your palms first to release aroma) and add two peach slices on the rim or inside the drink as garnish. Serve immediately with a straw.