Shave the skin off of the carrots with a vegetable peeler. Trim the carrots down to 6 inches so they fit nicely on a bun. If your carrots are very thick, consider shaving them down to the thickness of a regular hot dog so they don’t overwhelm the bun.
Fill a large pot with 2 inches of water and bring to a boil. Place the carrots in a steamer basket and lower the basket into the pot. Cover with a lid and steam the carrots for 10-15 minutes, or until fork tender.
Combine the vegetable broth, vinegar, soy sauce, mustard, smoked paprika, and garlic in a gallon Ziplock bag. Swish the marinade around in the bag until all of the ingredients are well combined.
When the carrots are finished steaming, add them to the Ziplock bag and spread them out in a single layer. You can put the bag of carrots into a shallow baking dish just in case it leaks. Refrigerate for at least 8 hours or overnight, turning the carrots a few times so all sides can evenly soak up the marinade.
Remove the carrots from the bag and discard the marinade. Heat a grill or nonstick pan to medium high heat. Place the carrots on the grill or in the pan and turn them so they get grill marks or sear marks on several sides.
Serve on whole wheat buns with desired toppings and enjoy!