Season the meat with salt and pepper and drizzle all over with EVOO.
Place a large Dutch oven over medium-high heat with two turns of the pan of EVOO. Add the pork to the pan and brown all over, about 4- 5 minutes on each side. Remove the pork to a platter and add the onion, celery, carrots, garlic and bay leaves to the pot and sauté for 3-5 minutes. Add the water to the pot (the liquid should come two-thirds of the way up the pork) and scrape the bottom of the pot to loosen the brown bits. Add the cilantro or parsley and oregano. Bring to a boil, then reduce the heat to simmer. Cover and cook for 2 1/2-3 hours, or until fork tender. Turn off the heat and let rest for 30 minutes. Remove the pork to a shallow platter and shred with forks; cover and reserve.