Trim the chicken thighs of excess fat and cut into 1/2- to 1-inch pieces; transfer to a bowl. Add the paprika, turmeric and salt, and use your hands to mix well. Let the chicken marinate at room temperature while you prepare the other ingredients, or cover and refrigerate overnight.
In a large pot, heat the oil over medium-high. Stir in the onions, lower the heat to medium-low and cook gently, stirring often to prevent scorching, until tender and translucent, 8 to 10 minutes. Add the garlic and continue to cook, stirring often, until most of the water from the onions has been cooked out and a glossy layer of oil has risen to the surface, about 5 minutes more.
Add the marinated chicken and stir to release the spices into the onion. Pour in the coconut milk and bring to a near boil. Let the coconut milk simmer briskly for about 4 minutes to thicken a bit. Lower the heat to medium-low and add the fish sauce. Stir in 1 1/2 cups water and bring to a near boil. The broth will thin out as the chicken starts to release its juices.
Lower to a gentle simmer and cook, stirring occasionally, until the chicken is tender, 30 to 45 minutes. Droplets of paprika-red oil will rise to the surface. Stir in the curry powder and cayenne, simmer briefly and remove from the heat.
If time permits, let the curry sit for at least 20 minutes before serving. This allows the chicken to soak in more flavors as the curry cools. Bring to a simmer again right before serving and taste, adding more salt or fish sauce if desired. Serve over rice or noodles, with bowls of cilantro and lime wedges.
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