Preheat an oven to 400 degrees F (200 degrees C). Lightly grease a baking dish.
Heat 2 tablespoons of olive oil in a large skillet over medium-high heat. Cook and stir the chicken and mushrooms until the chicken breasts in the hot oil until the chicken is no longer pink in the center and the juices run clear, about 15 minutes. Season with salt, pepper, and garlic powder and remove from heat. Stir in the zucchini, sweet potatoes, red potatoes, bread crumbs, Parmesan cheese, and 1/4 cup of olive oil; transfer to the prepared baking dish.
Bake in the preheated oven, stirring every 15 minutes, until potatoes are tender and easily pierced with a fork about 1 hour.
