Combine sugar and 1 cup water in a small saucepan over high heat; bring to a boil, stirring until sugar dissolves. Place pan into a large, ice-filled bowl, and let stand 20 minutes or until syrup is chilled, stirring occasionally.
Coarsely chop 5 peach halves; set aside remaining peach half. Combine chopped peaches, chilled syrup, lime juice, and salt in a blender, and process until smooth.
Thinly slice reserved peach half. Divide gazpacho evenly among 4 bowls; top with peach slices, Salted Candied Almonds, and mint, if desired. Serve immediately.