Bring a large pot of water to a boil. Add in the carrots and cook for about 7 to 8 minutes, or until just fork tender. Drain and rinse well with cold water.
In a small bowl, whisk together the soy sauce, vinegar, maple syrup, olive oil, garlic and smoked paprika. Transfer the mixture to a sealable gallon-sized plastic bag along with the cooked carrots. Mix to coat, then transfer to the refrigerator, keeping the carrots in a single layer. Marinate overnight or up to 24 hours.
When ready to eat, set a grill pan over medium high heat and allow it to heat up for several minutes. Place the carrots on the heated grill, shaking off the excess marinade first, and cook for about 5 to 6 minutes, rotating the carrots once or twice. Remove to a plate.
For the mango avocado salsa:
In a medium bowl, mix together the mango, red onion, cilantro, avocado, jalapeno, lime juice, olive oil, salt and pepper. Taste and adjust any of the flavors as desired.
Place one of the carrots in a bun. Top with some red cabbage and mango salsa. Serve immediately.
Prep time does not include time for the carrots to marinate overnight.