Chill a rocks glass with ice. Stir whiskey, syrup, and bitters with ice in a separate mixing glass for ~30 seconds. Dump the ice from the rocks glass and rinse with absinthe (pour, swirl, dump). Strain the stirred drink into the prepared glass. Express lemon peel over the top and discard.
New Orleans, mid-1800s. The first 'cocktail' to be called a cocktail by name. Pre-Prohibition rye, post-Prohibition absinthe ban substitutes (Herbsaint), now back to real absinthe.
Neat in a chilled rocks glass. No ice in the final drink. No garnish in the glass, just the expression. Serve with a small water back if asked.
Spicy, anise-forward, dry. Should not taste sweet. Should not taste boozy hot. The rinse is olfactory, not flavor.
Costing is directional. Confirm against current beverage grid when invoices update.