Heat oil in a large skillet over medium-high heat and brown chicken pieces 3–4 minutes; transfer to a plate.
Sauté onion until soft, 4–5 minutes, then add garlic and ginger for 1 minute.
Stir in curry powder, cumin, and turmeric; toast briefly until fragrant.
Add squash, diced tomatoes, coconut milk, and broth; bring to a simmer.
Return chicken to pan, cover, and cook 15–18 minutes until squash is tender and chicken is cooked through.
Season with salt and pepper, garnish with cilantro, and serve over rice or with naan.