Carefully remove the cabbage leaves one by one with a small sharp knife, taking care not to break them. Rinse in cold water, then boil in a large pot of boiling salted water in batches for just a few minutes, until softened. Gently remove the leaves and put the cooking water aside for later. Don't discard the tough outer leaves or torn ones because they will be used later to line the cooking pot, and the smaller inner leaves can be used for any patching up. You will need about 21 good leaves. Rinse these under cold water, shake off the excess, then lay them flat on trays lined with clean dish cloths. Cut away any thick bottom stems if they look like they won't roll up easily.
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.
Recipe courtesy of From Food from Many Greek Kitchens by Tessa Kiros/Andrews McMeel Publishing