Preheat the oven to 350F. Grease an 8-inch cake pan or other large deep round baking dish with the butter.
Heat a heavy pan with a little olive oil. Saute the carrot, onion, and celery until soft. Add the beef and pork and cook until the meat is browned. Add the garlic, bay leaf, thyme and white wine or cognac. Cook until aromatic. Add the tomatoes and cook until there’s very little liquid left. Salt and pepper to taste.
Remove from the stove and allow the mixture to cool enough so that it wouldn’t cook the egg when it’s added. Discard the thyme stems and bay leaf, then incorporate the rice and egg.
Lay the largest best cabbage leaf on the bottom of the pan. This will be your design when you flip it out of the pan at the end. Then begin laying other leaves around the edge so that they fall over the edge (you’ll be folding these back over the cake) but also make sure each added leaf overlaps the main large leaf. Once you have a layer, add a layer of the filling mixture and press down to flatten well. Top with another layer of cabbage leaves. Repeat until you’ve filled up the pan or have finished using all of the leaves or mixture. We typically get 3-4 layers. Finally, fold the outer leaves that are hanging over the pan back into the center. Make sure to tuck the edges of the last leaf into the pan. You will probably have some leaves left over, especially the small inside ones.
Cook for about an hour or until the cabbage has fallen and the cake is bubbling. Remove from the oven carefully. Let it cool a bit, then invert it on a plate to unmold it.