To make the Hatch chile barbecue sauce, heat the butter in a skillet on medium-low and add the onions. While occasionally stirring, cook for 5 minutes until fragrant then stir in the garlic. Cook for 1 more minute then remove from the heat.
Place the onions and garlic into a blender along with the Hatch chiles, vinegar, mustard, oil, honey, ketchup, salt, cumin, and cayenne. Blend until smooth then taste and adjust seasonings if needed.
To make the pork, place the meat in a Dutch oven, and toss with 2 tablespoons of the Hatch barbecue sauce. Evenly season the meat with the salt, pepper, smoked paprika, garlic powder, and Worcestershire sauce. Add enough water to the pot to almost reach the top of the meat, without covering it.
Place in the oven and cook uncovered until fork-tender, 2 to 2.5 hours.
When the meat is done, remove the pot from the oven. When cool enough to handle shred the meat, stirring in some of the pan juices to taste, if desired.
Toss the meat with ¼ cup of the sauce, then serve on buns with pickles, onions, and coleslaw with more sauce on the side.
Notes
While you may use any coleslaw that you prefer, I have found that 4 cups of shredded cabbage, 2 tablespoons of the Hatch chile barbecue sauce, 2 tablespoons of mayonnaise, and sliced green onions, along with salt and pepper to taste, makes a fine coleslaw and topping.