In a large pot, combine the onion, garlic, chicken or turkey, salt, and a pinch of pepper. Pour in the enchilada sauce, salsa verde, and 3 cups of water. Stir in the brown rice or white rice. Place the pot over high heat and bring to a boil, reduce the heat to medium-low, cover, and simmer, stirring occasionally, until the rice is cooked through and the chicken shreds easily, about 45 minutes. (About 18 minutes if using white rice). Remove the pot from the heat.