Season chicken with salt, pepper, lime juice, and half the oil. Rest 30 minutes.
Grill over medium heat for 5 to 6 minutes per side until lightly marked but not fully cooked. Transfer to a tray.
Make the Sauce:
Heat the remaining oil in a shallow skillet. Cook onion for 8 minutes, then add garlic and ginger for 1 minute. Add tomatoes, curry powder, turmeric, cumin, and cayenne and cook 8 minutes.
Blend the tomato mixture with coconut milk and water until smooth. Return it to the skillet and bring to a gentle simmer.
Finish the Chicken:
Add the grilled chicken and simmer gently, partly covered, for 25 to 30 minutes until the thickest pieces reach 165 F (74 C). Turn once and avoid a hard boil.
Stir in half the cilantro, adjust salt, and finish with lime wedges and the remaining cilantro.
Notes
A gentle simmer keeps the coconut sauce smooth and prevents the chicken from toughening.
Refrigerate for up to three days and reheat slowly; add a splash of water if the sauce thickens.