For those new to Peychaud's bitters, it is a bright red herbal bitters with more sharp flavors than Angostura (which tends to be more rooty) with a distinctive clove character.
A great history of the Sazerac Cocktail is provided here.
I find the standard 1 tsp. sugar recipe a tad too sweet. Also, I find Old Overholt a bit lacking in character, resulting in the same overall impression in a Sazerac cocktail. Based on the results of the rye tasting, my brand of choice for this cocktail is Wild Turkey Rye (101 proof). If you want something around 90 proof, I think that Sazerac Rye edges out Russell's Reserve Rye for use in this drink (although Russell's edged out Sazerac on a stand alone basis).
Although absinthe is recommended, so little is used that I find it hard to get too caught up with this aspect of the cocktail. Herbsaint is traditional as it locally produced in New Orleans. I typically use Granier pastis, our "house brand" (Julie is a big fan of pastis). Finally, I like to discard the lemon peel. While it may feel a tad undergarnished, I enjoy the purity of the resulting cocktail, with the aroma and essential oils of the lemon coating the rim and surface of the liquid.
The Setup (not shown - Hawthorne strainer)
2 oz. rye whiskey (Wild Turkey or Sazerac 6y)
1/2 tsp sugar
1/2 Tbsp branch water (southern for bottled water)
4-5 dashes Peychaud's bitters
Absinthe, Herbsaint or Pastis
Fill one old-fashioned glass with ice. In a second glass, place sugar and bottled (or otherwise neutral) water and swirl until dissolved. Fill with ice and add rye. Dump ice from the first glass and pour a tiny amount of absinthe (or substitute) and swirl to coat the glass. Discard excess. Stir the rye and ice mixture and strain into the absinthe-coated glass. Twist lemon peel over the drink's surface and rim the glass. Discard peel.