Heat the olive oil over medium high heat. Add the garlic, ginger, and onion. Cook for 4-5 minutes. Add the curry powder (or paste) and cook for 30 seconds. Stir in the chicken broth and turn off heat. Stir in the coconut milk.
Place the chicken and vegetables in the slow cooker. If you like your vegetables more tender crisp, wait and add the broccoli, snow peas, zucchini, and spinach during the last thirty minutes of cooking.
Pour the coconut curry mixture over top. Cook on low for 3-4 hours. Chop or shred chicken. Season with salt and pepper as needed. Serve with fresh lime juice. Cooking time may vary based on your slow cooker.