Heat a large skillet over medium-high heat. Cook and stir ground beef, onion, and garlic in the hot skillet until meat is browned and crumbly, 5 to 7 minutes. Drain and discard grease. Add carrots and mushrooms. Add tomatoes and break up using a fork. Add basil, sugar, 1 teaspoon salt, thyme, and nutmeg. Reduce heat and simmer meat sauce for 30 minutes.

Meanwhile, bring a large pot of water to a boil. Immerse cabbage in the boiling water and carefully break off leaves. Cook until leaves are limp, about 3 minutes. Drain.

Combine cottage cheese, 1/2 cup Parmesan cheese, parsley, 1 1/2 teaspoons salt, and oregano in a bowl and set aside.

Preheat the oven to 350 degrees F (175 degrees C). Coat a 9x13-inch baking pan with cooking spray and place a layer of cooked cabbage leaves flat in the bottom.

Reserve 1/4 cup of meat sauce. Layer 1/2 the remaining sauce on top of the cabbage leaves in the baking pan. Layer 1/2 the cottage cheese mixture followed by 1/2 the mozzarella cheese. Add a second layer of cabbage leaves, followed by remaining 1/2 of meat sauce, remaining cottage cheese mixture, and mozzarella cheese. Sprinkle reserved 1/4 cup meat sauce on top and finish with 1/2 cup Parmesan cheese.

Bake in the preheated oven for 45 minutes. Remove from the oven and let sit 15 minutes before slicing.