One way to distinguish between Cajun and Creole jambalaya: Traditional Cajun cooking doesn't include tomatoes. It does contain plenty of thyme, paprika, and oregano for a flavorful dinner.
Heat 2 tablespoons oil in a large enameled cast-iron skillet over medium-high. Season chicken with salt and pepper. Add chicken to skillet. Cook, turning occasionally to brown chicken all over, about 8 minutes. Remove chicken to a plate, leaving oil in skillet.
Add sausage, cook turning once to brown, about 1 minute per side. Remove to plate with chicken, leaving oil in skillet. Reduce heat to medium. Add remaining tablespoon oil, onion, celery, bell pepper, garlic, and 1 teaspoon salt. Cook, scraping brown bits from bottom of pan and stirring occasionally, until very soft and golden, about 8 minutes.
Add rice and cook, stirring constantly, 1 minute. Add bay leaf, paprika, cayenne, oregano, and thyme and continue to cook, stirring constantly, 1 minute. Return meat and any accumulated juices to skillet. Stir in broth, increase heat, and bring to a boil. Reduce to a simmer, cover, and cook until rice is tender, 20 minutes. Serve.