Rinse the dried peaches under cold running water, rubbing off any dust; drain well.
Sprinkle 0.5 cup granulated sugar evenly in a wide pot and cook over medium heat without stirring until melted and deep amber, 3–5 minutes; swirl the pot as needed.
Carefully add 12 cups water at once; it will sputter. Stir until any seized caramel dissolves and return to a bare simmer.
Add cinnamon sticks and whole cloves, then the rinsed fruit. Adjust heat to maintain a gentle simmer.
Cook until the peaches are plump to the pit and the liquid is a warm amber color, 35–45 minutes.
Stir in the remaining sugar to taste and simmer 2 minutes to dissolve fully.
Remove from heat; cool 30 minutes. Strain out and discard the whole spices. Transfer to a pitcher with the peaches, cover, and chill until very cold, 2–3 hours.
To serve, place 1 peach in each glass, fill with the chilled drink, and add ice cubes as desired.