How to Make It
In a food processor or blend, blend the basil, 1/4 cup of the olive oil, the garlic, pine nuts, Parmigiano-Reggiano, salt and pepper, until smooth, about 30 seconds; scrape down the sides as needed. Set aside.
Pound the chicken breasts to an even thickness. Brush each side with olive oil and sprinkle with a pinch of salt and a few turns of pepper. Grill the chicken breasts over moderately high heat until browned on the outside and cooked through, about 4 minutes on each side. Transfer to a cutting board and chop into 3/4-inch pieces.
Place the quinoa, water, and a pinch of salt in a saucepan and bring to a boil over moderately high heat. Reduce the heat to moderately low and simmer, stirring occasionally, until the quinoa is tender and most of the liquid has been absorbed, about 20 minutes.
Transfer the quinoa to a large boil, add the chicken, and then fold in the pesto until evenly distributed.