Bring 3 cups water to a boil in a saucepan. Stir in rice, 1 teaspoon salt, and butter. Cover, reduce heat to low, and cook 15 to 17 minutes, until water is absorbed. Set aside.
Heat oil in a cast-iron skillet over medium-high heat. Add onions and garlic; reduce heat to medium. Cook, stirring occasionally, until brown, about 10 minutes. Add chicken livers; cook, stirring, 5 minutes. Add rice and all remaining ingredients. Cook, tossing and stirring, until hot, 2 to 3 minutes. Serve immediately.