Pack a chilled coupe glass with ice to chill while you mix.
In a cocktail shaker, add the gin, Cocchi Americano, Cointreau, and freshly squeezed lemon juice. Fill the shaker with ice and shake hard until very cold and well diluted, 10–12 seconds, until the shaker is frosty.
Discard the ice from the coupe. Add the absinthe to the glass, swirl to coat the interior, then discard the excess.
Double-strain the cocktail into the absinthe-rinsed coupe.
Express the orange peel over the surface to release oils, then place it in the drink or discard, as preferred. Serve immediately.