Season the pork shoulder liberally on all sides with salt and pepper. Set aside.
Add 1 tablespoon vegetable oil to a Dutch oven over medium heat. Once the oil is hot, add chicken livers and cook to medium (150°F) and bouncy to the touch, about 8-10 minutes, flipping halfway through cooking.
Remove the livers from the pot and set aside. Add the remaining tablespoon of oil to the pan and add the seasoned pork shoulder, fat side down. Cook 4-5 minutes on one side until golden brown. Quickly add the chopped onion, bell pepper, celery, and smashed garlic to the pan around the pork, then add the water. The water should cover the pork shoulder by an inch. Using a wooden spoon, stir, taking care to scrape the bottom to release all the delicious brown bits. Add 1 tablespoon no-salt Cajun seasoning and 1 tablespoon kosher salt.
Increase the heat to high and allow the mixture to come to a boil. Then reduce to medium and cover with a lid.
Cook until the liquid is fragrant and greatly reduced and an inserted thermometer reads 200°F in the center of the pork, about 70-75 minutes.
Once cooked through, remove the pork shoulder onto a cutting board and allow it to cool for 20 minutes.
Meanwhile, scoop out 2 ½ cups of aromatics into a large measuring cup and set aside.
Roughly chop the pork shoulder into 2-inch pieces, being sure to leave most of the fat intact. Discard any gristle.
Add the cooked livers and 1 tablespoon braising liquid to the bowl of your food processor and pulse 4-5 times until it’s a rough paste. Remove from the processor and add to a large bowl.
Next, add the aromatics and rice to the food processor and pulse 3 times until the vegetables are broken down and the rice is just beginning to break down.
Add the pieces of pork and pulse 2 times until the pork is roughly chopped (but not emulsified) and the mixture looks moist. Place the mixture into the bowl with the livers.
Add remaining 2 teaspoons of no-salt Cajun seasoning, smoked paprika, cayenne pepper, parsley, chopped celery leaves (if using), and scallions.
Using a spatula, gently fold the mixture together until everything is evenly distributed. The mixture should look moist and hold together. If it looks dry, add the braising liquid one tablespoon at a time. Be careful not to add too much liquid, as the mixture won’t hold together. Taste for seasoning and adjust as needed.
Spread the mixture in an even layer in the bowl. Place plastic wrap directly on the surface of the filling, then cover the entire bowl with another layer of plastic wrap. Place in the refrigerator for at least four hours, or ideally overnight. This allows the mixture to chill completely and firm up so it’s scoopable and for the flavors to marry.
Once the filling is firm, prepare a 3-plate dredge setup: the all-purpose flour goes in the first plate, the beaten eggs into the second plate, then the breadcrumbs into the third plate. Season the flour with ½ teaspoon kosher salt.
Remove the filling from the refrigerator and scoop into two tablespoon portions (or use a #40 scoop) and roll gently into a ball. Roll the balls into the flour, then the egg, then the breadcrumbs, shaking off the excess after each ingredient. Set onto a parchment-lined sheet pan and place in the refrigerator until the oil is ready.
When ready to fry, add 2 inches of vegetable oil to a heavy bottom pot and heat over medium until the oil reaches 350°F. Working in batches, gently lower about 6-7 balls into the hot oil. Cook for about 3 minutes until golden brown all over. Remove from the oil and drain on a paper-towel lined plate. Allow the oil to return to 350°F before frying the remaining batches of balls.
Once all the balls are cooked, place on a serving plate and garnish with sliced scallions. Serve hot.