Season the duck pieces with salt and pepper.
Heat the vegetable oil in a large heavy pot over medium-high heat. Add the duck, skin side down, and sear for 5 minutes. Turn and sear on the second side for 3 minutes. Remove from the pot and drain on paper towels.
Add the sausage to the fat in the pot and cook, stirring, until browned, about 5 minutes. Add the onions, bell peppers, celery, salt, cayenne, 1 tablespoon of the Essence, and black pepper and cook, stirring often, until the vegetables are softened, about 5 minutes. Add the tomatoes, garlic, and bay leaves and cook, stirring, until the tomatoes give off some of their juices, about 2 minutes.
Add the thyme, stock, and duck. Bring to a boil. Reduce the heat to medium-low, cover, and simmer, stirring occasionally for 50 minutes.
Remove duck pieces from the jambalaya and cool slightly. Skim the fat. Discard skin and bones and shred duck meat.
Add the rice and bring back up to a simmer, cover and cook until the rice is barely tender, about 10 minutes.
Return the duck meat to the mixture. Season the shrimp with the remaining 1 tablespoon Essence. Add the shrimp to the pot, return to a simmer, and cover. Remove the pot from the heat and let sit, covered, for 15 minutes.
Add the green onions and parsley to the jambalaya and stir gently. Remove and discard the bay leaves. Adjust the salt, pepper, and cayenne to taste. Serve directly from the pot.