Muddle the sugar and bitters to a paste in an old-fashioned glass. Fill the glass with ice, add the bourbon and stir for 12 to 15 seconds with a bar spoon or long-handled spoon.
In a second old-fashioned glass, add the absinthe, swirl to coat the inside of the glass, then discard the excess absinthe. Strain the bourbon mixture into the absinthe-rinsed glass.
Serve without ice, neat. Garnish with a lemon peel.