In a large, heavy Dutch oven, over medium heat, melt 3 tablespoons of the butter. Meanwhile, using a paper towel completely dry the pork. The dryer the pork, the better, so you develop the gorgeous golden brown crust when you brown it. Season the pork with generously with salt and pepper. I love to "make it rain" salt, so probably about 3-4 good tablespoons.
When the butter starts to foam, add the pork and cook it about 4-6 minutes per side, until all the sides are a dark golden brown.
Once the pork is browned, and set aside on the plate, add the onions to the pot. Sauté, stirring often, until they are translucent, about 2-3 minutes. If you wantt to take this a step further and caramelize them, you'll add even more flavor to the dish. If you caramelize the onions, add more butter, turn your stove to medium low, and sauté until they're a deep golden brown. Caramelizing should take about 30-45 minutes. If you're short on time, you can sauté until translucent (like I mention above.)
Pour the cider over the onions, and bring the mixture to a boil to deglaze the pan (which means you're getting all those flavor bits left from the pork incorporated into the onions and cider). This should only take a couple of minutes.
Reduce heat to the lowest setting, add the pork, along with any juices, back to the pot. Sprinkle the thyme, cover and cook until the pork is easily shredded with a fork, about 1 1/2 hours - 2 hours. You want no resistance at all when you try to shred it! If the liquid starts reducing too much, and the meat starts sticking, add about 1/2 cup more cider.