Heat 1 Tbsp. olive oil in a pan over medium heat. Place the carrots in the pan and sauté for about 5 minutes. Add the barbecue sauce to the carrots and stir to combine. Allow the carrots to cook for about 15 minutes, until tender. Add a few tablespoons of water if the pan looks dry.
Meanwhile, in a medium sized bowl, whisk together the remaining olive oil, apple cider vinegar, lemon juice, parsley, salt and pepper. Add the cabbage to the bowl and toss with the dressing. Let the cabbage marinate until ready to serve.
Assemble the sandwich by placing a portion of the barbecue carrots on the bottom of each bun and topping with cabbage slaw.