In a small saucepan, warm the duck stock. In a small bowl, combine the egg white and cold water, then stir that mixture into the warm stock. Bring to a simmer. Remove pan from heat and let stand for 5 minutes. (As the egg white cooks, it will coagulate and trap fine particles from the stock.) Pour the stock through cheesecloth into a small, clean bowl to strain out the particles and egg white. Add the salt and smoked pepper.
Pour the seasoned stock, Génépy and absinthe into a rocks glass.
Garnish with toasted juniper berries, whole caraway seeds and a rosemary sprig.