Coat a deep 12- to 14-inch non-stick skillet or sauté pan with non-stick cooking spray and place over MEDIUM heat. Add the ground beef and cook, stirring to break up the meat, until cooked through and no longer pink, about 5 to 6 minutes. Drain off the excess grease and transfer the beef to a plate and set aside.
Return the empty skillet to the heat and add 1 tablespoon olive oil. Add the onion and celery and cook, stirring occasionally, until the onion is translucent and celery is tender, about 5 minutes. Add the garlic and cook, stirring, for an additional 2 minutes. Return the beef to the skillet and season with ½ teaspoon salt, basil, thyme, oregano, and black pepper. Add the uncooked rice, marinara sauce, petite diced tomatoes, beef broth, and parsley, and stir well to combine. Bring the mixture to a low boil then reduce the heat, cover and simmer for about 25 to 30 minutes, or until most of the liquid has been absorbed and the rice is tender (taste to check).
Preheat oven to 350 degrees F. Coat a 13- x 9-inch baking dish with non-stick cooking spray.
While meat mixture is cooking, add remaining 1 tablespoon of olive oil to a Dutch oven and place over MEDIUM heat. When oil is hot add the cabbage and season with remaining ½ teaspoon salt and black pepper, to taste. Cook, stirring occasionally, for about 5 minutes or just until cabbage is softened and fork tender. The cabbage will begin to brown but if it begins to burn, lower the heat under the pot. Remove the pot from the heat.
Scatter half of the cabbage across the bottom of the prepared baking dish. Spoon half of the meat mixture over the top and sprinkle with half the mozzarella and Parmesan. Repeat layers with the remaining cabbage and meat mixture. Reserve remaining cheese for later.
Cover the baking dish with foil and transfer to the preheated oven. Bake for 25 minutes. Remove the foil, sprinkle the remaining mozzarella and Parmesan cheese over the top, and continue to bake for an additional 15 minutes until cheese is melted and lightly browned. Remove from the oven and allow the casserole to sit for at least 5 minutes before serving.