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In a bowl, mix together the pork, beef, rice (uncooked), onion, tomatoes, parsley, paprika and juice of 1 lemon, and season well with salt and a few grinds of pepper. Spoon a healthy wooden spoonful (about 1 3/4 ounces) of mixture onto the lower middle of each leaf. Start rolling up the filling from the bottom. Make one full turn, tuck both sides toward the middle, then continue rolling. Don't roll too tightly, as the rice will expand during cooking, but they must be compact to hold their bundles.
Heat half the butter and half the oil in a large wide nonstick pot that has a lid. Line the bottom with the reserved outer cabbage leaves. Arrange the cabbage rolls tightly on top in concentric circles, starting from the outside. When one layer is full, make another on top until all the rolls are in. Dot the rest of the butter on top and drizzle the remaining oil over. Sprinkle lightly with salt and pepper. Pour in enough of the reserved cabbage water to just cover, about 3 cups. Find a plate that fits into the pot snugly and put it on top, inverted, to keep the bundles from moving around. Cover, bring gently to a boil and simmer for about 40 minutes, or until the rolls are tender and the rice is cooked. Add 11/2 cups of water a few minutes before the end of cooking, as you will need it for the sauce.
Whisk the eggs, the juice of the remaining 3 lemons and a little salt and pepper in a bowl. Whisk in about a cup of the hot cooking liquid to acclimatize the egg, remove the pot from the heat, then pour the egg mixture into the pot. Distribute it by rocking the pot well. Return the pot to the lowest heat, shaking and rocking it until the sauce thickens a little. Be careful not to overheat it, as the eggs will scramble. If it doesn't look saucy enough, add a little hot water. Remove from the heat and let rest, covered, for a few minutes, rocking the pot now and then. Serve the rolls hot or warm, drizzled with a good amount of sauce, a light sprinkling of salt if necessary and a grinding of pepper.