Add pork to slow cooker and cook on high for 3-4 hours. Remove, shred, and return to slow cooker to keep warm.
Cut broccoli into smaller, bite-sized pieces, then place into a microwave-safe dish along with water. Cook on high for 2-3 minutes or until crisp-tender.
Mince garlic; zest and juice half the lime.
Add garlic, lime zest, lime juice, salt, and pepper to a small mixing bowl. Slowly add oil while whisking continuously until well blended.
Peel carrot and grate into lime mixture. Add steamed broccoli and mix well.
Dice avocado and mash with the back of a fork. Scrape into a small bowl and add zest and juice from the remaining half of the lime; mix well.
Lay out pitas and spread avocado mixture onto each. Top with an even amount of the veggie mixture, then some shredded pork and lettuce.
Serve pitas with grapes on the side.