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Fill an Old Fashioned glass completely with ice and let it chill.
In a mixing glass, place 1 sugar cube (or ¼ ounce simple syrup) and add 3 dashes Peychaud's Bitters; crush the sugar cube (or stir simple syrup) into the bitters until dissolved.
Pour 2 ounces rye whiskey into the mixing glass.
Fill the mixing glass two-thirds full with ice and stir for about 30 seconds to chill and dilute.
Discard the ice from the chilled Old Fashioned glass, pour in ¼ teaspoon absinthe, swirl to coat, then discard any excess.
Strain the stirred rye mixture into the absinthe-rinsed glass without adding fresh ice.
Twist a lemon peel over the drink to express oils, rub the rim with the peel, then drop it in as garnish.