Heat 2 tablespoons of the oil in a large skillet or Dutch oven over medium-high heat until shimmering. Season the chicken with salt and pepper, then add to the skillet. Cook, turning occasionally, until golden brown on all sides, about 8 minutes (chicken will finish cooking later). Transfer the chicken to a plate.
Add the sliced sausage to the skillet and cook until browned and slightly crisp around the edges, about 2 minutes. Remove to the plate with the chicken.
Add the remaining 1 tablespoon of oil to the skillet. Stir in the onion, celery, bell pepper, and garlic. Season lightly with salt and cook, stirring occasionally, until the vegetables are softened and golden, about 8 minutes.
Stir in the rice and cook, stirring constantly, for 1 minute to lightly toast.
Add the bay leaf, paprika, cayenne pepper, oregano, and thyme. Stir for another minute to bloom the spices.
Return the chicken and sausage to the skillet, along with any juices from the plate.
Pour in the chicken broth, stirring to combine. Bring the mixture to a boil.
Reduce the heat to low, cover, and simmer until the rice is tender and the liquid is absorbed, about 20 minutes.
Remove from heat and let the jambalaya rest, covered, for 5 minutes. Discard the bay leaf. Taste and adjust seasoning as needed.
Garnish with sliced green onions and/or hot sauce, if desired, and serve hot.