Sprinkle chicken with curry powder, season with salt and pepper. In a large skillet, heat 1 tablespoon oil over medium-low. Add chicken and cook until golden and opaque throughout, 30 to 35 minutes, turning once.
In a blender, combine 1 tablespoon oil, cilantro, ginger, lime juice, vinegar, sugar, and 3 tablespoons water. Puree until smooth; set aside.
In a medium saucepan, heat remaining teaspoon oil over medium. Add shallot and turmeric; season with salt and pepper. Cook, stirring occasionally, until shallot begins to soften, 1 to 2 minutes. Add coconut milk and 1 cup water; bring to a boil. Stir in rice; return to a boil. Reduce to a simmer, cover, and cook just until rice is tender, 12 to 14 minutes. Remove pan from heat and let stand, covered, for 5 minutes. Fluff with a fork just before serving. Serve rice topped with chicken and drizzled with sauce.