Heat the oil in a 5- to 6-quart Dutch oven over medium-high heat. Season the pork with the oregano, cumin, ¾ teaspoon salt, and ½ teaspoon black pepper. Add the pork to the Dutch oven and cook, turning, until brown on all sides, 6 to 8 minutes; transfer to a plate.
Add the onion, garlic, and cayenne to the Dutch oven and cook, stirring, until the onion is softened, 3 to 5 minutes. Return the pork to the Dutch oven and add the broth. Bring to a simmer and cook, covered, stirring occasionally, until the pork is very tender and shreds easily, 1½ to 1¾ hours.
Transfer the pork to a medium bowl and, using 2 forks, shred the meat; return it to the Dutch oven and stir into the cooking liquid. (Alternatively, the pork can be made in a slow cooker: Combine the pork, oregano, cumin, ¾ teaspoon salt, ½ teaspoon black pepper, onion, garlic, and cayenne in a 4- to 6-quart slow cooker. Add ½ cup broth and cook until very tender, on low for 7 to 8 hours or on high for 4 to 5 hours.)