Grease the slow cooker with olive oil, then evenly arrange the shredded cabbage on the bottom. Place the boneless chicken breasts on top of the cabbage and drizzle them with lime juice.
In a small bowl, stir the smoked paprika, ground cumin, chili powder, fine sea salt, and ground black pepper. Sprinkle the seasoning mixture on both sides of the chicken, then pour any extra seasoning into the pot over the cabbage. Top with the minced garlic.
Secure the lid and cook on low for 3 to 4 hours. The chicken should reach an internal temperature of 165ºF in 3 hours, but it can be cooked longer if needed.
Shred the chicken with two forks and toss it with the shredded cabbage directly in the slow cooker.
Serve warm in taco shells or lettuce leaves with your favorite toppings. Leftover shredded chicken can be stored in an airtight container in the fridge for 3 to 4 days.
