For the garnish: In a medium saucepan over high heat, combine 1 cup water with 1/2 cup sugar. Bring the mixture to a boil. Using a candy thermometer to monitor the temperature, continue boiling until 230 degrees F. Add the orange and/or lemon peels along with a splash triple sec. Lower temperature to medium heat. Turn the peels occasionally, until the liquid is reduced to a thin syrup, about 15 to 20 minutes. Reduce the heat to low and simmer until the syrup is thick and the slices are tender but still intact, turning occasionally, about 10 minutes. Transfer the orange or lemon peels to a cooling rack set over a baking sheet to cool.
For the cocktail: Mix sugar, water, and bitters in an old-fashioned glass. Drop in the cherry and a candied orange and lemon peel. Muddle into a paste using a muddler or the back end of a spoon. Pour in the bourbon, fill with ice cubes, and stir. Garnish with additional orange or lemon peel.