Preheat your oven to 400°F (200°C).
In a small, deep baking dish, whisk together the olive oil, vegetable broth, soy sauce, maple syrup, paprika, garlic powder, and liquid smoke. Season with a large pinch of Kosher salt and adjust to taste. Place the carrots in the dish with the marinade and toss to coat them evenly.
Bake in the preheated oven for 20-25 minutes, or until the carrots are tender and slightly caramelized, turning them halfway through baking.
While the carrots are baking, prepare the buns. Slice each dinner roll in half horizontally, then cut a vertical slit down the middle of each half, going about ¾ of the way through to create a pocket for the carrot dog. Lightly toast the mini buns if desired. Once the carrots are done, place each roasted carrot in a bun.
Serve the mini hot dogs with mustard, ketchup, or any other toppings of your choice.