Place the chicken breasts in a bowl, add the Italian dressing. Mix well, cover and marinate in a fridge for at least 30 minutes or overnight and up to 24 hours.
Preheat oven to 375F (for skin-on breasts) or 350F for skinless breasts.
Transfer the chicken breasts and the marinade into a deep baking dish. Bake until the internal meat temperature reaches 160F, about 35-40 minutes at 375F (or 50-60 minutes at 325F). If you are using skin-on breasts, you may want to use the broiler mode in the last few minutes to crisp up the skins.
Meanwhile, prepare the balsamic mushrooms. Heat the oil in a wok or a cast iron pan over medium-high heat. Add the mushrooms and the onions and stir-fry until nicely caramelized, about 7-8 minutes. Add the pepper and the balsamic vinegar, then add salt to taste. Continue cooking for another minute. Remove from heat and set aside.
Remove the baking dish with the chicken from the oven. Let the chicken rest for 5 minutes before serving.
Garnish with fresh basil leaves and serve with balsamic mushrooms on the side.