Fill a mixing glass with ice and stir briefly to chill.
With one pull from from the dropper of a 1 oz bitter bottle, carefully coat the inside of a chilled coupe glass with about 1/6 tsp absinthe, allowing the excess absinthe to remain in the bottom of the glass.
Strain excess water from mixing glass, adding fresh ice if needed, then add the cognac, syrup and bitters to the mixing glass.
Stir for 45 full rotations, then strain with a Hawthorne strainer into the prepared coupe.
To prepare the garnish, use a vegetable peeler to create a 2- to 3-inch-long and 1- to 1-1/2-inch-wide peel from a room-temperature lemon.
Express the lemon oils over the drink, paint the rim of the glass with the peel, and touch the stem of the glass once or twice with the peel.
Using a paring knife or kitchen shears, neatly trim the edges of the peel into an isosceles acute triangle, then cut a 1-inch slit in the center of the peel about 1/8 inch from the base of the triangle.
Starting from the tip of the triangle, roll the peel at an angle slightly off axis into a barrel shape.
Affix the rolled peel to the lip of the couple and release, which will cause the peel to unfurl into a swan-like shape.
*Demerara syrup: Combine equal parts demerara sugar and water in a saucepan. Heat and stir until the sugar dissolves. Let cool before using.