Preheat an oven to 375 degrees F (190 degrees C). Grease a 9x13 inch baking dish.
Loosen the skin on each chicken breast to form a pocket, and set aside.
Heat the butter in a skillet over medium heat, and cook and stir the onion, parsley, and basil until the onion is translucent, about 5 minutes. Stir in the zucchini, and cook and stir until tender, about 2 more minutes. Remove the skillet from the heat, and thoroughly mix the torn bread, egg, Swiss cheese, salt, and pepper into the zucchini mixture until well-combined.
Stuff the chicken breasts with the zucchini mixture, place them in the prepared baking dish, and bake in the preheated oven until the chicken breasts are golden brown and the stuffing is hot, 50 to 60 minutes.