Preheat the oven to 425°. In a blender, puree the tomatoes until fairly smooth. Strain through a fine sieve into a saucepan, pressing on the solids; you should have about 2 1/2 cups of puree. Add 3 thyme sprigs and simmer over moderate heat, scraping down the sides, until thick and glossy, 15 minutes. Season with salt and pepper; discard the thyme.
Meanwhile, in a flameproof roasting pan, toss the carrots with 3 tablespoons of the oil; season with salt and pepper. Cook over high heat until lightly browned, 3 minutes. Roast the carrots in the oven for 40 minutes, turning, until browned and very tender. Halve the carrots lengthwise. Leave the oven on.
Rinse the tapioca. In a large saucepan of boiling water, cook the tapioca over high heat, stirring, until al dente with a small white core at the centers, 17 minutes. Drain and rinse the tapioca and stir into the tomato puree.
Heat the remaining 3 tablespoons of oil in a large ovenproof skillet until nearly smoking. Season the chicken with salt and pepper and add to the skillet, skin side down, with the remaining 3 thyme sprigs. Cook over high heat just until the skin is lightly browned, 5 minutes, then roast in the oven for 15 minutes, until cooked through; discard the thyme.
Preheat the broiler. Broil the chicken, skin side up, until golden and crispy, 3 minutes. Arrange the chicken and carrots on plates. Spoon off the fat from the pan juices. Add the juices to the tapioca and spoon it around the chicken. Top with the pistachios and serve.