Bring sugar, water, and peaches to a boil in a small saucepan. Simmer for 15 minutes. Use a fork to press down on the peaches and extract their flavor. Let cool completely. Pour mixture through a fin-meshed sieve into a bowl or liquid measuring cup. Press down on peaches with a spoon to get as much liquid out as possible. Discard solids.
Combine the bourbon, 1/2 to 1 ounce of peach simple syrup, and the bitters in an old fashioned glass with ice and serve.