In a bowl, season the chicken with 1 tablespoon olive oil, salt, pepper, and cajun seasoning. Let sit for 15 minutes to allow the seasonings penetrate.
In a separate bowl, season shrimp with a little bit of olive oil, plus salt and pepper to taste. Place back in the refrigerator until it is time to cook them.
Heat 1 tablespoon olive oil in a large Dutch oven over medium heat. Add the chicken and sauté until all sides are golden brown. Add the sausage and cook until lightly browned on both sides. Transfer chicken and sausage into a bowl and set aside. The chicken will not be fully cooked at this point, but will finish cooking later.
To the same pot, add the onions, green bell pepper, celery, and minced garlic. Cook until tender, stirring occasionally, about 5 minutes.
Add thyme, old bay seasoning, cayenne pepper, and salt and pepper to taste (about 1/2 teaspoon salt and 1/4 teaspoon black pepper). Add tomato paste and stir to coat.
Stir in crushed tomatoes, chicken, and sausage. Cook, covered, for about 2 minutes.
Stir in the rice, and add the chicken broth, mixing to combine. Add the bay leaf. Bring to a boil. Reduce heat to low and simmer, covered, for about 20 minutes.
Then layer the shrimp on top, cover, and cook for another 10 minutes, or until rice is tender and liquid is absorbed. Stir the jambalaya to fully combine. It will be somewhat wet.
Remove from heat, stir slightly, and let sit for about 15 more minutes before serving to let to flavors and liquid further absorb. Serve and top with thinly sliced green onions.