Fill a small rocks glass with ice to chill and add the absinthe. When the glass is chilled, stir, then discard the ice. Pour out the absinthe, but first swirl it around to make sure the interior of the glass is coated.
In a cocktail shaker, combine the Cognac, simple syrup and bitters. Vigorously shake with ice, then strain into the prepared rocks glass. Run the rim of the glass with the lemon twist, then serve the cocktail straight up, garnished with the lemon twist.