Cut the pork shoulder into several large pieces, if needed, to make browning easier.
Heat the vegetable oil in a large Dutch oven over high heat.
Season the pork shoulder all over with salt and then place in Dutch oven.
Cook all sides until brown—about 10 minutes
Add the onion, garlic, lime juice, spices and chicken broth. Bring to a boil then reduce to medium-low, cover and simmer until pork is very tender—about 2 ½ hours.
Preheat oven to 400ºF.
Transfer the pork to a large baking sheet, saving the juices. Drizzle the pork with some of the juices and lightly season with salt. Bake the pork in the oven until browned, about 30 minutes. Drizzle more juices on meat about every 10 minutes and use forks to shred meat as it browns.