In a large enameled cast-iron casserole, heat the oil until shimmering. Add the onion, garlic, ginger and chiles and cook over moderate heat, stirring occasionally, until softened, about 5 minutes. Stir in the curry powder and turmeric and cook for 2 minutes. Add the broth and coconut milk and season with salt and pepper. Bring to a boil and simmer for 5 minutes. Add the potatoes and cook over moderate heat until barely tender, 5 minutes. Add the cauliflower and squash, cover partially and cook over moderately low heat, stirring occasionally, until the vegetables are tender, 15 minutes.
Add the chicken to the casserole, season with salt and pepper and cook just until white throughout, about 5 minutes. Serve over rice, with lemon wedges.