Pour the vegetable oil into the bottom of the slow cooker. Put the pork roast into the slow cooker and pour in the barbeque sauce, apple cider vinegar, and chicken broth. Stir in brown sugar, mustard, Worcestershire sauce, chili powder, onion, garlic, and thyme. Cover and cook on high for 5-6 hours or until the roast shreds easily with a fork.
Take the roast out of the slow cooker, and shred using two forks. Place the shredded meat back into the slow cooker and toss with the juices.
To make the coleslaw- In a large bowl combine shredded cabbage and carrots. In a small bowl combine mayonnaise, sugar, and vinegar. Mix well, and pour over the cabbage mixture and toss to coat.
Heat a large skillet to medium heat. Spread the inside of both halves of the sliders buns with butter. Place buttered side down into the skillet and toast until golden brown. Spoon pulled pork onto the bottom half of a slider bun, spoon on coleslaw with a slotted spoon, and place on the top half of the bun.