Combine mayonnaise and ranch dressing mix in a gallon-sized resealable plastic bag to make the marinade. Place chicken in the bag, squish around to cover, and refrigerate while preparing the other ingredients.
Preheat oven to 400 degrees F (200 degrees C). Grease a large rimmed baking sheet the size of your oven with olive oil.
Place red potatoes and baby carrots along the outer edges of the baking sheet, leaving room in the middle for the chicken. Season with salt and pepper.
Pour breadcrumbs into a bowl or onto a plate. Remove chicken from marinade and drain off excess marinade. Coat chicken in the breadcrumbs and place on the baking sheet with some room in between the pieces.
Bake in the preheated oven until chicken is no longer pink in the center and the juices run clear, 25 to 30 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Turn on the oven's broiler and broil for an additional 3 to 5 minutes.