Generously season the chicken with salt and pepper on both sides. Place the chicken in a 1-gallon resealable plastic bag.
Pour in 1/2 cup Balsamic vinaigrette and seal bag, removing as much air as possible. Lay on a rimmed baking sheet to catch any leaks and refrigerate for 1 hour.
Preheat oven to 400 degrees. Line a rimmed baking sheet with aluminum foil.
Place the vegetables on a rimmed baking sheet. Toss with 1/4 cup Balsamic vinaigrette until evenly coated. Arrange the vegetables in a single layer, around the edges of the prepared baking dish. Sprinkle vegetables with salt and pepper.
Remove chicken from ziplock and discard marinade. Nestle chicken pieces into the center of vegetables. Season both sides of chicken generously with salt and pepper.
Bake in preheated oven and bake chicken, flipping halfway through, about 16-18 minutes (Mine only needed 16 minutes.)
Sprinkle parmesan cheese over chicken and vegetables and broil until chicken is golden, 2-5 minutes (depending on how close the chicken is to the heating element) and a digital thermometer registers 165 degrees when inserted into the thickest part of the chicken breast.