Pat the pork dry and season all over with salt and pepper.
Heat 2 tablespoons oil in a large pot over medium-high heat. Add the pork, in batches if necessary, and cook without moving for 4-5 minutes per side until well-browned.
Cover with 8 cups water and add the quartered onion, garlic, a few shakes more of salt and pepper, paprika, and the oregano. Bring to a boil and then reduce heat and simmer for 3 hours.
After the pork has simmered for 3 hours, use tongs to transfer the pork to a large bowl and shred it into large pieces.
Pour the broth through a fine-mesh sieve into a bowl and discard all the solids. Reserve the broth.
Return the pot back to the stove.