For the chicken: Heat the oil in a heavy, 12-inch nonstick skillet over medium-high heat. Sprinkle the chicken with salt and pepper. Add the chicken and pepper flakes to the skillet and cook until the outside of the chicken is white, stirring frequently, about 5 minutes. Using a slotted spoon, transfer the chicken to a plate. Add the onion, bell peppers, and rosemary to the skillet and cook until beginning to brown, stirring frequently, about 7 minutes. Add the dry vermouth and bring to a boil, stirring up any browned bits. Return the chicken to the skillet with any juices on the plate. Add the broth, lemon zest and juice, and tomato paste to the skillet. Bring the liquid to a boil. Cover, reduce the heat and simmer until the chicken is tender, about 25 minutes.

Meanwhile, prepare the quinoa: Place the quinoa in a heavy small saucepan. Cover generously with cold water, swish around a few times, and then pour through a fine strainer. Return the quinoa to the pan and repeat rinsing and draining 3 more times. Return the quinoa to the pan, add 1½ cups water and salt. Bring to a boil. Reduce the heat to low, cover, and cook until the water is absorbed, about 15 minutes. Remove the pan from the heat and let stand 5 minutes to steam. Fluff the quinoa with a fork. Mix in the oil and basil. Season to taste with salt and pepper.

When the chicken is tender, uncover and simmer until the liquid is syrupy, about 10 minutes. Season to taste with salt and pepper. Divide the quinoa among 4 warmed plates. Spoon the chicken, vegetables, and sauce over and serve.