Heat a large heavy pot or Dutch oven over medium-high heat and add a splash of Vegetable Oil. Pat the Chicken Thighs dry, season lightly with Kosher Salt and Black Pepper, then brown the thighs on both sides until golden (about 4–5 minutes per side). Transfer the chicken to a plate and set aside.
Slice the Andouille Sausage and brown it in the pot for 3–4 minutes until it releases its fat and develops color; remove and set aside with the chicken.
Reduce the heat to medium, add a little more Vegetable Oil if needed, then add the diced Onion, Green Bell Pepper and Celery. Sauté until the vegetables are softened and aromatic, about 5–7 minutes. Stir in the minced Garlic and cook for another 30–60 seconds.
Add the Long-Grain White Rice to the pot and stir to coat the grains in the fat and vegetables, toasting the rice lightly for 1–2 minutes.
Pour in the Canned Diced Tomatoes with their juices and the Chicken Stock. Return the browned Chicken Thighs and Andouille Sausage to the pot. Season the mixture with Sweet Paprika, Dried Thyme, Dried Oregano, a Bay Leaf, a pinch of Cayenne Pepper (to taste), plus more Kosher Salt and Black Pepper as needed. Stir to combine.
Bring the pot to a gentle boil, then reduce the heat to low, cover, and simmer until the rice is almost tender and has absorbed most of the liquid, about 18–22 minutes (timing will vary by rice). Avoid stirring too often to allow the rice to cook evenly.
When the rice is nearly done, nestle the Shrimp into the surface of the jambalaya, cover, and cook until the shrimp turn pink and opaque, about 4–6 minutes. Taste and adjust seasoning with additional Kosher Salt, Black Pepper or Cayenne Pepper if desired.
Discard the Bay Leaf, fluff the jambalaya gently with a fork to combine the meats and rice, and let it rest off the heat for a few minutes to meld the flavors.
Garnish with sliced Green Onions and chopped Parsley before serving warm. Enjoy straight from the pot for a rustic, satisfying meal.
