Trim the tops and bottoms off the onions, then peel them. Carefully score a single slit down one side of each onion, being careful not to cut all the way through the layers. Boil the onions in salted water for 10 to 15 minutes until pliable. Reserve 4 cups of the onion broth, then drain and cool the onions. Gently peel apart the layers, keeping each piece intact for stuffing.

Warm the olive oil in a skillet over medium heat and cook the chopped onion until softened, about 8 minutes. Add the garlic and cook until fragrant. Add the ground beef, season with salt and pepper, and cook until no longer pink. Stir in 1 cup of the tomato puree and cook until slightly thickened. Add the rice, 1/2 cup of the reserved onion broth, and a little extra salt. Simmer for 6 to 8 minutes until the liquid is absorbed. Do not overcook, as the rice will continue to soften in the oven. Remove from heat and fold in the parsley, oregano, mint, and dill. Set aside.

Place a heaping spoonful of the filling in the center of one onion layer. Roll the onion around the filling to form a neat packet. Repeat with the remaining onion layers and filling.

Arrange the stuffed onions seam-side down in a baking dish, packing them snugly in a single layer so they stay rolled during baking.

Whisk together the remaining 2 cups of tomato puree, 2 cups of reserved onion broth, balsamic vinegar, and salt and pepper to taste. Pour the sauce over the stuffed onions. Cover the dish with foil or parchment paper.

Preheat the oven to 400°F (200°C). Bake the covered dish for 45 minutes, then uncover and bake for an additional 35 to 45 minutes, until the onions are tender and lightly caramelized. Switch the oven to broil for the final 5 minutes to add golden color on top.

Let the stuffed onions rest for a few minutes before serving. Garnish with extra chopped parsley and serve with crusty bread, yogurt, feta, or a fresh salad on the side.