Season roast with salt and pepper. Warm vegetable oil in a large skillet over medium-high heat. Cook pork, turning often, until browned on all sides, about 15 minutes. Transfer to slow cooker.
Pour off all but 2 Tbsp. fat in skillet. Add chopped onion, minced garlic, and seeded and chopped Anaheim chiles, poblanos and jalapeño. Sauté until softened, about 5 minutes. Pour in chicken broth; bring to a boil, stirring to pick up browned bits. Stir in barbecue sauce and Dijon mustard. Pour over pork in slow cooker. Cover; cook on low until pork is very tender, 6 to 8 hours.
Transfer pork to a platter; cool. Skim fat from cooking liquid, transfer to a pan and cook over medium-high heat until reduced to 2 1/2 to 3 cups. Pull pork into shreds, discarding fat and gristle. Return pork to cooker, pour liquid over and stir. Season with salt and pepper and serve.
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