Combine carrot, onion, and wine in slow cooker.
Sprinkle roast on all sides with chili powder, rubbing it into meat. Season on all sides with salt and add to slow cooker. Cover and cook on low until roast is falling-apart tender, about 6 hours.
Use tongs or a slotted spoon to transfer meat to a large bowl. Use two forks to shred meat into bite-sized pieces. Moisten/season with cooking juices to taste. Cover to keep warm and set aside.
In a medium bowl, combine cabbage and mayonnaise. Season with salt and pepper to taste.
Arrange 2 tortillas on each serving plate. Fill tortillas with pork, cabbage mixture, and avocado and serve. (Alternately, arrange pork, cabbage mixture, and avocado on a platter and let guests make their own tacos.)
Cooking tip: For even more flavorful carnitas, brown roast on all sides in a lightly oiled skillet before sprinkling with chili powder.