Bring a large pot of water to a boil.

Heat the oil in a medium pan and sauté the onions over medium heat, stirring occasionally, until browned and caramelized, about 20-30 minutes.

While the onions cook, remove the core from the cabbage. Fill a large pot with water, place the cabbage in the pot, and boil until the leaves soften and start to peel away from the head, about 15-20 minutes.

Bring 4 cups of water to boil in a small saucepan. Stir the rice into the boiling water. As soon as the water returns to a boil, strain and rinse the rice. Set aside.

Remove the cabbage from the pot and let cool slightly until it can be handled. Carefully, peel the leaves off by first shaving the base of the stem vertically with a paring knife to remove the hard triangular rib. Reserve the trimmings. Return the cabbage to the pot for a few minutes to soften more if you find that the inner leaves difficult to remove. Make sure you have 12-14 leaves.

Make the filling: In a medium bowl, add the ground beef, rice, and ¾ of the caramelized onions, and mix well to combine. Add the eggs, ½ of the can of tomato paste, the juice from ½ a lemon, sugar, salt, and pepper, and mix to combine.

To assemble: Lay a cabbage leaf flat and, depending on its size, place ½-¾ cup of the filling near the rib end of the leaf. Fold the left and right sides in, then fold the base of the leaf up over the filling, then roll tightly towards the top outer edge of the leaf to form a rectangular package. Place the wrapped leaf seam-side down on a platter while you prepare the remaining leaves. Continue with the rest of the leaves until all of the filling has been used up.

To a large pot add the remaining onion, any trimmings from the cabbage stems, the remaining tomato paste, the juice from ½ a lemon, the crushed tomatoes, 1 tablespoon of salt, and 1 teaspoon of pepper, and bring to a boil. Layer the stuffed cabbages over the the sauce and top with any remaining loose cabbage leaves. Add water to just cover the contents of the pot. Bring to a boil, then cover and reduce to a simmer. Cook for 1 hour, then taste and adjust for seasoning, adding additional lemon juice, sugar, salt, or pepper as needed.

Remove the lid and cook until sauce reaches desired thickness, about 15-20 minutes.

Serve immediately with sauce drizzled on top. Cabbage rolls can be stored an airtight container in the refrigerator for 5 days.