Heat the oil in a large skillet over medium heat. Add the onion, celery, and poblano and cook until soft, 3 to 5 minutes. Add the garlic and cook until fragrant, 45 to 60 seconds. Add the chicken livers and cook until no longer pink, about 3 minutes. Stir in the rice, thyme, paprika, cayenne pepper, the 2 cups water, and chopped collard greens. Season with 1/2 teaspoon of coarse salt and freshly ground black pepper. Increase the heat to medium-high, and bring to a boil. Cover and decrease the heat to simmer. Cook until the rice is tender, about 18 minutes, or for brown rice, about 35 minutes.
Remove from the heat and let rest, covered, for an additional 5 minutes. No peeking! Add the green onions and fluff with a fork to combine. Taste and adjust for seasoning with salt and pepper. Serve warm with hot sauce on the side.