Pat the beef shank pieces dry with paper towels. Season all the beef pieces generously on all sides with about half the kosher salt and the black pepper.
Place your large stockpot or Dutch oven on the stove over high heat. Add 12 cups (2.8 liters) of cold water to the pot. Add the seasoned beef shank pieces and beef chuck to the cold water. Add the halved white onion, peeled garlic cloves, bay leaves, and the remaining salt to the pot. Bring the pot to a full rolling boil over high heat, which will take about 12 to 15 minutes.
Once the water reaches a boil, you'll see a brownish-gray foam rising to the surface. Use your fine mesh skimmer to scoop this foam off the surface and discard it. Work quickly but thoroughly. Keep skimming every few minutes for the first 10 to 15 minutes of boiling until the foam production slows significantly and the broth begins to run clear.
Once the foam has been skimmed, reduce the heat to low. You want a gentle, steady simmer, with just a few small bubbles breaking the surface at a time. Add the dried Mexican oregano to the pot. Place the lid on the pot, leaving it slightly ajar to allow a little steam to escape. Let the beef simmer for 2 to 2.5 hours, or until the meat on the beef shank is very tender and beginning to pull away from the bone. Check the pot every 30 minutes. If the liquid level drops significantly, add an additional cup of hot water to maintain the level. Taste the broth at the 90-minute mark and adjust salt as needed.
Using tongs, carefully remove the beef shank pieces from the pot and transfer them to a cutting board. Remove and discard the bay leaves, onion halves, and garlic cloves from the broth with a slotted spoon. Allow the beef shank to cool for about 5 to 10 minutes, until it is comfortable enough to handle. Pull the meat away from the bones with your fingers or two forks. The meat should separate very easily at this point. Discard the bones. Break any large pieces of meat into bite-sized chunks and return all the meat to the pot. If using beef chuck, it can remain in the pot as is since it has been simmering the whole time.
Stir the tomato paste into the broth until it fully dissolves. Add the roughly chopped Roma tomatoes and the jalapeño pepper (whole or halved depending on your heat preference). Bring the broth back up to a gentle simmer over medium heat.
Add the corn rounds, carrot chunks, and potato quarters to the pot. Stir gently, then bring the soup back to a steady simmer. Cook for 15 to 20 minutes, until the carrots begin to soften at the edges and the potatoes are starting to cook through but are not fully tender yet.
Add the zucchini half-moons to the pot. Nestle the cabbage wedges into the liquid. Add the tied bundle of fresh cilantro, pushing it down into the broth. Stir gently to combine everything. Continue simmering for another 15 to 20 minutes, until the potatoes are fork-tender, the zucchini is just cooked through, and the cabbage has wilted and softened. Remove and discard the cilantro bundle and the jalapeño (if you prefer not to serve it).
Taste the broth one final time and adjust the salt and pepper as needed. If the broth tastes a little flat after all the vegetables have been added, squeeze in the juice of half a lime directly into the pot. Ladle the soup into large, deep bowls, making sure every bowl gets a generous portion of broth, beef, corn, potatoes, carrots, zucchini, and cabbage. Serve immediately with lime wedges, chopped white onion, fresh cilantro, and warm corn tortillas on the side.