Grease the slow cooker with olive oil, then evenly arrange the shredded cabbage on the bottom. Place the chicken breasts on top of the cabbage and drizzle them with lime juice.
In a small bowl, stir the smoked paprika, cumin, chili powder, 1 ½ teaspoons fine sea salt, and a ¼ teaspoon 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. (Or use a ½ teaspoon garlic powder, instead.)
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.
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Notes
Nutrition information is for roughly 6 ounces of cooked chicken with cabbage. (A quarter of this whole batch.) This information is automatically calculated and is just an estimate, not a guarantee.Update Note: This recipe was updated in January 2025 to increase the spices slightly. I doubled the paprika and chili powder and added ground black pepper.