Place an old-fashioned glass in the freezer or fill it with ice water to chill.
In a separate mixing glass, muddle the sugar cube with 2 dashes of Peychaud's bitters until dissolved.
Add the rye whiskey and a large handful of ice to the mixing glass.
Stir for 20–30 seconds until well chilled and properly diluted.
Discard the ice (or ice water) from the chilled old-fashioned glass.
Add the absinthe to the chilled glass, swirl to coat the interior, then discard any excess.
Strain the stirred cocktail into the absinthe-rinsed glass.
Express a lemon zest over the surface to release its oils, then run it around the rim and place it on top to garnish.
The traditional Sazerac was originally made with Cognac, but a mid-19th century rye whiskey substitution - driven by the phylloxera epidemic that devastated French vineyards - became the standard. Some bartenders split the base 25ml rye / 25ml Cognac to honor both versions.
My Bar Shelf reads your shelf and shows you every cocktail you can make from what you already own, including Sazerac.