In a second Old-Fashioned glass, muddle sugar with a few drops of water. Add several small ice cubes, the rye whiskey, Peychaud's bitters and the Angostura bitters. Stir well.
Remove ice from chilled Old-Fashioned glass and swish around the absinthe until its inside is thoroughly coated, pouring off the excess.
Strain cocktail mix into chilled Old-Fashioned glass and garnish with a twist of lemon peel.