In a medium saucepan, mix together the broth, vinegars, soy sauce, liquid smoke, and spices. Bring to a simmer and continue to cook for 5 minutes.
Meanwhile, toss the oil and carrots together in a small roasting pan.
Pour the hot marinade over the carrots. Cover the pan with aluminum foil and let it sit for at least 30 minutes and, preferably, up to 1 1/2 hours to let the flavors infuse the carrots.
Preheat oven to 425°F (218°C).
Slide the covered pan in the oven and roast for 20 minutes. Take off the foil, stir the carrots and flip them over to coat them in the marinade again, and roast until tender on the outside with a little bit of resistance in the center when you stab them with a fork, 20 to 25 minutes more, depending on the size of the carrots.
Slip each carrot hot dog in a bun and add your choice of toppings.