Melt butter over low heat in lg. pot. Add sausage. Pour 1/4 c. red wine over sausage. Cover and let cook, turning occastionally.
Place ground beef in med. bowl. Break apart with clean hands. Add egg and 1/4 c. Parmesan and 1 tsp. oregano and gently work through. Do not overwork. Make golfball-size meatballs by taking sm. amount of meat mixture and gently tossing from hand to hand.
Make filling. In lg. bowl, combine ricotta and remaining Parmesan. Stir in 3 minced garlic cloves, Italian parsley, and remaining oregano. Fold in mozzarella pearls.
Season steak with salt, pepper, and dried oregano on both sides. One by one, starting at short end, spread cheese mixture over 1/3 of ea. steak piece. Starting at filling end, roll up steak. Secure with butcher's twine. Tuck in ends.
Using tongs, gently add braciole to pot. Add tomatoes, tomato paste, carrot shavings, and 1/2 of diced onion. Add 1/2 c. red wine. Season with salt. Cook at least four hours; six is better. Stir occasionally. If boiling down too much, reduce heat and add 1/2 c. water. Add more wine to taste.
One hour before serving, add basil and remaining onion, garlic, and fresh oregano. S/P to taste.
Serve with freshly grated Parmesan, as desired.