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, 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 rice and cook, stirring, for 2 minutes.
Add the thyme, stock, and duck. Bring
to a boil. Reduce the heat to medium-low, cover, and simmer, stirring
occasionally, until the rice is tender, about 30 minutes.
Season the shrimp with the remaining 1
tablespoon Essence. Add the shrimp to the pot and cook until they turn pink,
about 5 minutes. 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.