How to Make It
In a medium saucepan, heat the oil. Add the lemongrass, ginger, garlic and cinnamon stick and cook over moderate heat, stirring occasionally, until the garlic is golden, about 3 minutes. Transfer to a heatproof medium bowl, add the vinegar and sambal oelek and let cool completely. Fold in the tomatoes and 2 teaspoons of salt and marinate at room temperature for at least 6 hours. Remove the cinnamon stick before serving.
In a medium saucepan, heat the oil. Add the onion and garlic, season with salt and cook over moderate heat, stirring occasionally, until softened and just starting to brown, about 10 minutes. Add the cilantro, ginger and curry paste and cook, stirring occasionally, until the ginger is softened, about 5 minutes. Add the stock, coconut milk and cream and bring to a boil. Reduce the heat to moderately low and simmer, stirring occasionally, until reduced to 2 cups, about 25 minutes.
Stir the lime juice, fish sauce, sugar and kaffir lime into the broth and remove from the heat. Let stand at room temperature for 20 minutes.
Light a grill or preheat a grill pan. Brush the chicken with oil and season with salt. Grill skin side down over moderate heat until lightly charred and the skin is crisp, about 7 minutes. Flip the chicken and grill until just cooked through, about 7 minutes longer. Transfer to a carving board and let rest for 5 minutes.
Strain the broth through a fine sieve and return it to the saucepan; discard the solids. Bring to a simmer and season with salt. Slice the chicken crosswise 1/2 inch thick and transfer to shallow bowls. Ladle the coconut broth around it and spoon some of the tomatoes and their marinade on top. Garnish with small basil leaves and serve.