Light a grill. Using kitchen shears, cut along both sides of the chicken backbone; discard the backbone. Turn the chicken breast side up and press down firmly on the breast bone to crack and flatten it. Using a sharp knife, cut deep slits to the bone 1/2 inch apart along the chicken legs and thighs. Transfer the flattened chicken to a medium baking dish.
In a small bowl, whisk the coconut milk with the curry paste and brown sugar until smooth. Rub the curry mixture all over the chicken, into the slits and under the skin; season with salt and pepper.
Grill the chicken skin side down over moderate heat until the skin is browned and crisp, about 10 minutes. Turn the chicken skin side up, cover and grill over moderate heat until cooked through, about 20 minutes. Transfer the chicken to a cutting board and let rest for 5 minutes. Carve the chicken and serve.