Heat a large Dutch oven over medium-high heat. Add oil and sausage and cook, stirring occasionally, until very well browned, about 10 minutes. Add the chicken cubes and continue to cook, stirring frequently, until the chicken is browned and cooked through, about 5 - 10 minutes. Add the onion and cook, stirring frequently, until soft and translucent, about 5 minutes.
Add the ham, garlic, thyme, bay leaves, black pepper and red pepper flakes and cook until fragrant, about 2 minutes. Add the white rice and stir to coat the grains with the rendered fat and spices. Continue to cook, stirring constantly, until the rice is translucent, then add the chicken broth, Kitchen Bouquet and kosher salt.
Bring the broth to a simmer while stirring constantly to prevent any rice from sticking to the bottom of the pot. Once the liquid begins to simmer, reduce the heat to low, cover, and cook until the rice is tender, about 25 minutes. Remove from the heat and stir to fluff the rice. Remove the bay leaves and season to taste with salt. Garnish with the scallions and hot sauce.