Place the beef chuck roast and beef broth in a slow cooker. Cook on low for 8 hours until tender and easily shreddable.
Remove the beef, discard excess fat, and shred with two forks. Stir in 1 cup of BBQ sauce and keep warm.
In a large bowl, whisk together the mayonnaise and apple cider vinegar. Add the shredded cabbage and carrots, tossing to combine. Chill until serving.
Preheat the oven to 400F. Toss the sliced zucchini with olive oil and the Italian herb blend. Spread on a baking sheet.
Roast the zucchini for 15 minutes until tender and slightly browned.
Assemble the sliders by placing a generous scoop of pulled beef on the bottom bun, adding a spoonful of coleslaw, and topping with the upper bun. Serve with the roasted zucchini on the side.