Rinse your glass with absinthe, swirling it to coat the inside of the glass. Pour out the excess.
In a mixing glass, add your sugar cube, Peychaud’s bitters, and Angostura bitters. Let the sugar absorb the bitters and then muddle it gently in the glass.
Add your rye whiskey and a handful of ice cubes to the mixing glass and stir gently for about 30 seconds. Strain the mixture from the mixing glass into your old-fashioned glass.
Twist a lemon peel over the drink, then drop it into the glass for garnish.