With a sharp knife, remove the core from cabbage. Place cabbage, core side down, in a large pot with about 2-3 inches of boiling salted water, cover and cook on low for 5 minutes. Remove from heat and carefully turn the cabbage cut side up. Next, using tongs, remove the cabbage leaves and set them aside to cool a bit. You will need about 10 leaves. Coarsely chop the remaining cabbage and place it over the bottom of a large pot or dutch oven (optional). Set aside.
Using a knife, cut away the thick center stem from the outside of each leaf (see photo above), without cutting all the way through.
Heat the butter in a large pan over medium heat until melted. Add the onion, garlic, bell pepper, and carrot and saute, stirring frequently, for 5 minutes or until soften. Remove from heat and let cool.
In a large bowl, combine the pork, chicken, rice, egg, sauteed onion mixture, salt, and pepper. Place 1/4 to 1/2 cup of meat on each cabbage leaf near stem end. Fold over each side and tightly roll the cabbage roll. At this point, cabbage rolls can be frozen until ready to use.
Once you have made all your rolls, place them (with the end of the leaf face down) on top of the chopped cabbage in the pot. Add peppercorns, chopped carrot and bay leaves. In a small bowl, combine tomato paste with 1 cup of water and pour over rolls. Add enough water so the rolls are almost covered. Add salt to taste. Cover the pot and bring to a boil over medium-high heat. Reduce heat and simmer, covered, for 50-60 minutes, or until the filling inside the cabbage is cooked through.
For the sauce: Remove the cabbage rolls from the pot. Melt 1 tablespoon of butter in the microwave and mix it with flour. Add this mixture to the pot, stir gently and boil for a few minutes. Take off the heat and stir in the sour cream.
Serve the cabbage rolls topped with the sauce and accompanied by boiled or mashed potatoes. Garnish with fresh dill, if desired. Enjoy!