Season the chicken breasts with salt and black pepper. Heat the olive oil in a deep, nonstick pan at moderate temperature.
Braise the chicken breasts on both sides for about 10 min. total. Take the chicken breasts out and braise the onions (chopped into thin crescents beforehand) for 2-3 min.
Return the chicken to the pan and cover the dish with a lid. Cook until the chicken is ready, about another 10-15 min. Sprinkle the rest of the basil over the dish at the end and serve.