Place the pork shoulder chunks in a large Dutch oven or stockpot with the halved onion, 6 garlic cloves, bay leaves, and 1 tablespoon of salt. Pour in the water or chicken broth to submerge the pork.
Bring to a boil over high heat, then reduce to a low simmer. Skim off any foam that rises during the first 15 minutes. Cover partially and simmer for 1 1/2 to 2 hours, until the pork is fork tender and easily shreds.
Remove the pork with tongs and set it aside to cool slightly. Discard the bay leaves and onion halves from the broth. Keep the broth warm on low heat.
While the pork simmers, bring a separate pot of water to a boil. Add the tomatillos, poblano peppers, and jalapeño peppers. Boil for 8 to 10 minutes until the tomatillos turn olive green and everything is soft. Drain and cool slightly.
In a dry skillet over medium heat, toast the pepitas for 3 to 4 minutes, stirring often, until they pop and turn golden. Let cool slightly, then set aside a small handful for garnish.
Add the boiled tomatillos, poblanos, jalapeños, toasted pepitas, cilantro, cumin, and Mexican oregano to a blender. Pour in 1 cup of the warm pork broth and blend on high for 1 to 2 minutes until completely smooth.
Heat the vegetable oil in a large skillet over medium-high heat. Carefully pour in the blended green sauce and cook, stirring often, for 8 to 10 minutes until it deepens in color and thickens slightly.
Pour the cooked green sauce into the pot with the reserved pork broth. Shred the cooled pork into bite-sized pieces and return it to the pot. Add the drained hominy and stir everything together.
Bring the pot to a gentle simmer and cook for 20 to 25 minutes, allowing the flavors to meld and the hominy to warm through. Taste and add more salt as needed.
Ladle the hot pozole verde into bowls, ensuring each portion gets plenty of hominy and pork. Top with shredded cabbage, sliced radishes, chopped onion, reserved toasted pepitas, and serve with lime wedges and crisp tostadas on the side.