Fill a rocks glass with ice and leave to chill.
In a mixing glass, put a sugar cube (or 7.5 ml simple syrup) and add three dashes of Peychaud's Bitters. Crush the sugar (or stir the syrup) into the bitters until it dissolves.
Add 60 ml rye whiskey to the mixing glass.
Fill the mixing glass about two-thirds full with ice, then stir for about 30 seconds to chill and dilute.
Discard the ice from the chilled rocks glass, pour in ¼ teaspoon absinthe and swirl to coat, then pour off any excess.
Strain the rye whiskey mixture into the absinthe-rinsed glass without adding fresh ice.
Twist a strip of lemon peel over the drink to release the oils, rub the rim with it, then drop it in as a garnish.