Season the chicken with 1 tsp. Cajun spice. In a large, heavy pot or Dutch oven over medium heat, add the oil. When the oil is hot, add the chicken in 2 batches and cook until just cooked through and lightly browned; set aside.
In the same pot, brown the sausage. Remove and set aside.
Add the onion, bell pepper, minced garlic, thyme and bay leaves in 2 Tbsp. of remaining oil. Season lightly with Cajun spice. Cook over medium heat, stirring occasionally, until the onions are golden brown, about 8–10 minutes.
Add the rice, stirring for a couple minutes until the rice is coated with oil and all is well mixed.
Return the chicken, sausage and the ham to the pot, add the stock along with a teaspoon Cajun spice, stir and bring to a simmer. Reduce the heat to low, cover, and cook until stock is absorbed, about 30 minutes.
Lightly season the shrimp and stir into the pot the last 10 minutes of cooking.
Fluff rice with a fork and taste for salt. Remove the bay leaves. Just before serving, stir in the scallions and parsley. Serve with hot sauce.