Chill an old-fashioned glass by filling it with ice and letting it sit while preparing the rest of the drink.
In a separate mixing glass, muddle the simple syrup and Peychaud bitters together
Add the rye whiskey and ice to the bitters mixture and stir.
Discard the ice in the chilled glass and rinse it with absinthe (or substitute) by pouring a small amount into the glass, swirling it around and discarding the liquid.
Strain the whiskey mixture from the mixing glass into the old fashioned glass.
Garnish with a lemon twist. Traditionalists will say that the lemon twist should be squeezed over the drink to release its essences but that the twist should not be dropped into the glass itself.
Served neat, never on the rocks