Set oven to 425˚F. In a bowl, combine vegetable stock, vinegar, soy sauce, honey, smoked paprika, and garlic. Whisk until ingredients are evenly distributed. Place carrots in an appropriately sized hotel pan and pour stock mixture over top, carrots should not be submerged. Allow to marinate 30 minutes - 1 hour, tossing occasionally.
Cover hotel pan with foil and roast carrots for 30 minutes. Remove foil and turn carrots to coat in the marinade. Return to oven and cook until tender. Hold warm for service.
Set grill to high. For each serving, place 1 carrot "hot dog" on grill to mark. Toast 1 bun.
Place grilled carrot in toasted bun and top with 1 tablespoon caramelized onions. Finish with a drizzle of yellow mustard and charred tomato ketchup. Serve warm.
Carrot hot dogs can be cooled and reheated for service as needed.
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