Cook rice according to package directions, being careful to not overcook. Allow to cool by placing in the refrigerator for 10-15 minutes before adding eggs in step 8.
Roughly chop onion and sauté in 1 tablespoon butter in a non-stick skillet until lightly caramelized. Remove onions to a separate bowl.
In the same skillet used to cook the onions, and in the remaining butter, sauté the red pepper until soft, about 2 to 3 minutes. Remove the pepper to a separate bowl.
In the same skillet, add the pulled, chopped pork, barbecue sauce, and liquid smoke. (Alternatively, one could use a prepared barbecue pork product from the supermarket or local barbecue restaurant, but be careful that it’s not too wet). Heat through and stir until all components are well distributed and excess moisture is removed. Turn off heat.
Add the onion, red bell pepper, and mild pepper rings to the pork mixture. Mix together well.
Prepare an ovenproof dish, standard 7 cup, 1.6 L size. Spray the bowl liberally with non-stick spray. Coat the dish with 1/8 to 1/4 cup dried bread crumbs by turning the dish so that the sides are well coated as well as the bottom. Reserve the remaining bread crumbs for the top.
Mix the cooked rice with 2 beaten eggs and 3 tablespoons dried bread crumbs.
Using a rubber spatula, line the bottom of the dish and sides with the rice mixture. Reserve about 1/3 for the top.
Layer the filling ingredients. 1/3 of the Mozzarella, 1/3 of the goat cheese, 1/2 of the barbecue pork mixture. 1/3 of the Mozzarella, 1/3 of the goat cheese, 1/2 of the barbecue pork mixture. End with 1/3 of the goat cheese and 1/3 of the Mozzarella.
Finish by covering the top with the remaining rice mixture, using a spatula to smooth evenly to the sides. Dust with remaining bread crumbs.
Bake in a 400°F oven for 30 minutes.
Allow to cool 10 minutes.
Run a knife along the sides to release.
Place a serving plate on top, invert and remove the Memphis Barbecue Mold.