In chilled cocktail shaker or pint glass, pour club soda over sugar cube. Using muddler or back of large spoon, gently crush sugar cube. Swirl glass until sugar dissolves, 20 to 30 seconds, then add bitters and Cognac and set aside.
Pour absinthe into chilled double old fashioned glass or stemless wineglass. Holding glass horizontally, roll between your thumb and forefinger so absinthe completely coats the interior, then discard excess.
Add ice to cocktail and stir until well chilled, about 20 seconds. Strain cocktail into chilled glass rinsed with absinthe. Using channel knife, cut thin 4-inch strip of peel from lemon directly over glass, then place peel in glass and serve.