Fill an old-fashioned glass with ice and set aside.
Place the sugar cube in a second old-fashioned glass and add the bidders. Muddle these together until the sugar cube is broken and thoroughly mixed with the bitters.
Add the rye whiskey and stir until the sugar dissolves.
Fill a small spray bottle with the absinthe. Discard the ice cubes and spray the interior of the chilled glass with absinthe.
Pour the whiskey mixture into the absinthe-coated glass and twist lemon peel over the cocktail. Serve immediately.