It was invented in New Orleans, I believe, by an apothecary named Peychaud. That’s why Peychaud’s Bitters are an important ingredient. Here is the recipe my brother gave me:

Chill an old fashioned glass in the freezer. Get it really really cold, thirty minutes ahead of time.

Put a sugarcube (or two) in a cocktail shaker and douse it with Peychaud’s Bitters. The cube should be saturated. I usually add a small splash of water.

With a cocktail muddler (or some blunt instrument), smash up that sugar, bitters and water until you’ve got a syrupy mixture in the bottom of the shaker.

Now add ice and two ounces of Rye Whiskey to the shaker. 

Pull the chilled glass out of the freezer and add enough Absinthe to coat the inside of the glass. Rotate the glass sideways so the Absinthe swirls around and covers the inside surface area. Since the glass is so cold, the Absinthe should thicken and adhere to the sides. Dump out any excess Absinthe in the sink.

Strain the cocktail shaker into the glass. Twist a lemon peel around the rim and drop it in the drink.