Place the chicken pieces in the slow cooker. Add the chicken broth, coconut milk, garlic, ginger, fish sauce, soy sauce, and chili flakes. Stir to combine.
Cover and cook on low for 5 hours, until the chicken is tender.
Add the broccoli florets and green beans to the slow cooker. Stir gently, cover, and cook for 45 more minutes until the vegetables are crisp-tender.
Stir in the spinach and Thai basil leaves until the spinach wilts. Serve the stew in bowls with lime wedges on the side.