In a saucepan, sauté the onions and garlic in the oil. Add the beef and cook over high heat while crumbling with the back of a wooden spoon. Once the beef is cooked and its juices have evaporated, add the oregano, chili powder, cheese, cream, broth, and chives. Cook and stir until smooth. Season with salt and pepper. Set aside.
Place a tortilla in a large pie plate. Cover with about 1/5 of the meat mixture. Sprinkle with 1/5 of the cheese. Repeat, alternating tortilla, meat, and cheese until all the ingredients have been used. Top with cheese.
Cover the tortilla cake with aluminum foil. Bake for 20 minutes. Remove the foil and continue cooking for about 20 minutes. Let stand for 5 minutes before cutting into wedges. Serve with salsa and sour cream, to taste.