If you’re using chicken pieces or the drumettes: in a large skillet with a lid over medium heat, heat 2 tablespoons of the oil until it shimmers. Add the chicken. Cover and cook for about 20 minutes, turning occasionally, to brown all sides. Remove the chicken and its juices to a shallow pan and keep warm. Omit this step if using shredded cooked chicken.
In a blender jar or work bowl of a food processor, combine the tomatoes, garlic, and ¼ cup water. Process for 20 seconds, or until smooth. Set aside.
In a large skillet over medium heat (use the same pan you used to cook the chicken, if applicable), combine the remaining 2 tablespoons oil and the rice. Cook, stirring every minute, until the rice turns a light golden yellow, about 5 minutes.
To the rice in the skillet, carefully stir in the processed tomato mixture, the onion, 2½ cups water, the tomato sauce, Holy Trinity, and salt. Stir well to combine.
If using bone-in chicken, crowd the pieces in a single layer over the rice, otherwise skip to the next step.
Return the skillet to high heat. Bring the liquid to a boil, then reduce the heat and simmer, covered, for 15 minutes.
Set aside off the heat, covered, for 5 minutes to finish cooking.
If using shredded chicken, stir the chicken into the rice to heat through just before serving.