Cook the meat: Heat the olive oil in a large skillet over medium heat. Add the ground pork and chicken livers. Cook until fully browned and cooked through, stirring occasionally.
Cool and process the meat: Transfer the cooked meat to a bowl and let it cool slightly. Once cooled, pulse it in a food processor for 30 seconds to 1 minute, or until it reaches a fine, crumbly texture. Do not over-process. Set aside.
Sauté the vegetables: Wipe out the skillet, then return it to the stove. Add the butter and melt over medium heat. Add the diced celery, green bell pepper, and onion. Cook for about 5 minutes, or until the vegetables are soft.
Add aromatics and spices: Stir in the minced garlic, Cajun seasoning, paprika, 1 teaspoon of salt, thyme, oregano, and cayenne. Cook for 30 seconds, just until fragrant.
Combine everything: Add the crumbled meat mixture back to the skillet. Stir in the chicken broth, parsley, and hot sauce. Bring to a gentle simmer, then turn off the heat.
Stir in the rice: Fold in the cooked white rice until fully incorporated. Let the mixture cool to room temperature.
Chill the mixture: Transfer the mixture to the refrigerator and chill for at least 30 minutes, or up to 2 hours, to firm it up.
Form the balls: Once chilled, use a cookie scoop or your hands to form the mixture into 2-inch balls. Place the formed balls on a plate or tray.
Chill again: Refrigerate the formed boudin balls for another 20 minutes to help them hold their shape during breading and frying.
Prepare the breading stations: In one shallow bowl, mix the flour with the remaining teaspoon of salt, garlic powder, and onion powder. In a second bowl, whisk the eggs with the buttermilk (and a dash of hot sauce, if desired). In a third bowl, add the breadcrumbs.
Bread the balls: Roll each chilled boudin ball in the flour mixture, then dip it in the egg wash, and finally coat it with breadcrumbs. Set the breaded balls aside.
Heat the oil: In a deep frying pan or Dutch oven, heat about 3 inches of vegetable oil to 375°F (190°C). Use a thermometer for accuracy. Fry the boudin balls: Working in batches, fry 5–6 balls at a time for about 5 minutes, turning halfway through, until golden brown and crispy. Let the oil return to temperature between batches.
Drain and cool: Transfer the fried boudin balls to a wire cooling rack set over a baking sheet to drain excess oil.
Serve: Serve hot with spicy Cajun remoulade sauce and a sprinkle of freshly chopped parsley, if desired.