In an Old-Fashioned glass (not a mixing glass; it's part of the ritual), muddle a sugar cube with a few drops of water.
Add several small ice cubes and the rye whiskey,* the Peychaud's bitters, and the Angostura bitters.
Stir well and strain into a second, chilled, Old-Fashioned glass in which you have rolled around a few drops of absinthe (no substitute really works, but you can try either a mix of Pernod and green Chartreuse, or Absente) until its inside is thoroughly coated, pouring off the excess.
Garnish with a twist of lemon peel