Grease the slow cooker with the olive oil, then arrange the shredded cabbage evenly on the bottom. Arrange the chicken breasts on top of the cabbage, and drizzle them with the lime juice.
In a small bowl, stir together the paprika, cumin, chili powder, and salt. Sprinkle the seasoning mixture on both sides of the chicken, and dump 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 stand being in there longer, if needed. Use two forks to shred the chicken, and toss it with the shredded cabbage directly in the slow cooker.
Serve warm in taco shells or lettuce leaves, with all of your favorite toppings. Leftover chicken can be stored in an airtight container in the fridge for up to 3 days.
Nutrition information is for the taco meat only. This information is automatically calculated, and is just an estimate, not a guarantee.You can use chicken thighs instead of breasts, if you prefer. They tend to stay more tender, without drying out in the slow cooker, so they can cook longer, if needed for your schedule.