Wash, peel, and trim your carrots so that they are relatively similar in thickness from end to end, and fit the bun. For a similar look to a hot dog, you can round the ends if you prefer.
Bring a large pot of water to a boil. Add salt to taste. Place medium carrots in the boiling water, cover, reduce heat to a rolling boil. Cooking time will vary. It could take anywhere from 10-15 minutes. Be careful not to overcook the carrots or they will fall apart. You want them to be fork tender, yet still firm.
While the carrots are cooking, prep the marinade. In a small bowl, whisk together the vegetable stock, apple cider vinegar, liquid smoke, mustard, and spices.
Drain the carrots, and rinse under cool water to stop the cooking.
Place the cooked carrots in a single layer in a shallow bowl. Pour the marinade overtop. Allow to marinate as long as possible anywhere from 1-2 hours.
Now, you're carrots are ready for the grill! Brush the carrots with olive oil. Heat your grill or grill pan over medium-high heat. Place on your heated grill or grill pan. Cook on each side for about 2 minutes, or until you get those gorgeous charcoal marks.
Place the carrot dogs in the bun, and top with all your favorite toppings! Enjoy!