Set a rack in the bottom half of the oven and preheat to 350°F.
Add the water, onion, salt, thyme, and thyme, and tomatoes to a large dutch oven or covered roasting pan, stir to combine, and then put the chicken - and the livers, giblets, and neck ,if using - into the pot; the chicken should be not quite covered with liquid. Cover and bake for 30 minutes, turn the bird over, and bake for another 30 minutes.
Add the black pepper, red pepper flakes, and turmeric to the pot, stirring it in around the bird, and flip the bird once again. Cook for 15 minutes; remove cover and cook 15 minutes longer, or until a thermometer inserted in the thigh reads 155°F.
Remove the pan from the oven and transfer the bird to a cutting board; reserve the broth in the pan. Discard the neck and giblets, if used, and tent the bird with foil.
In a 4 quart saucepan, heat the olive oil over medium heat, and when it shimmers, add the dry rice, stirring it to coat. Cook, stirring often, for about 3 minutes, until toasted. Add 4 1/2 cups of the broth from the roasting pan, cover, and cook over low heat until the liquid is absorbed and a crust has developed on the bottom of the pan, 25 to 30 minutes. The surface of the rice may appear damp, but the bottom should be dry. Scrape the bottom layer up and fold into the rest of the rice, cover, remove from heat, and allow to continue steaming until ready to serve.
Season the remaining cooking liquid to taste with salt, then heat over medium high on the stovetop. Once it simmer, whisk in the flour paste and cook for 5 to 7 minutes, stirring occasionally, until it is a thick gravy.
To serve, loosely mound the rice in a deep serving platter, set carved sections of the bird on top of the rice, and drizzle the rice and bird with gravy.