Rinse the rice under cool water until mostly clear. Drain well.
Warm the olive oil and butter in a heavy pot over medium-high heat.
Season the chicken with a pinch of salt and pepper. Brown in batches until golden.
Transfer browned chicken to a bowl. Leave the drippings in the pot.
Sauté the chopped livers, if using, until lightly browned. Scrape up browned bits.
Reduce heat to medium. Cook onion, bell pepper, and celery until softened.
Stir in garlic, Cajun seasoning, paprika, and thyme. Cook until fragrant, about 30 seconds.
Add the rice and toast, stirring, for 1 to 2 minutes.
Return chicken and any juices to the pot. Pour in broth and add bay leaves.
Bring to a boil. Cover, reduce heat to low, and simmer 18 to 20 minutes.
Remove from heat and rest, covered, 5 minutes. Fluff, fold in scallions and parsley, and season to taste.