Serve any leftover roasted vegetables as garnish. For a healthier wrap, omit the mayonnaise.
Heat oven to 450 degrees. In a small bowl, combine the shredded chicken and mayonnaise, if using, and set aside. Drizzle the head of garlic and beet with 1 tablespoon olive oil for each, and wrap each separately in foil. Place in the oven to cook until tender, 45 to 60 minutes. Remove from oven, and let cool. When cool enough to handle, peel beet, and cut into 1/2-inch pieces; set aside. Squeeze garlic from skins; set aside.
Place the sweet potato, parsnip, 3 tablespoons olive oil, 2 teaspoons salt, and 1/4 teaspoon pepper in a medium bowl, and toss to combine. Place the mixture in a large roasting pan in the oven, and roast, turning vegetables as necessary, until vegetables are starting to soften, 15 to 20 minutes.
Meanwhile place zucchini, yellow pepper, red pepper, beans, apple, 1 tablespoon olive oil, 1 teaspoon salt, and 1/4 teaspoon pepper in a medium bowl, and toss to combine. Add to the partly roasted vegetables, and continue to roast until vegetables are tender, about 15 minutes more. Remove from oven, and set aside to cool to room temperature.
Melt the butter in a medium skillet set over medium-low heat. Add the Vidalia onion, and cook, stirring occasionally, until the onion is very soft, about 10 minutes. Remove from heat, and set aside.
Spread a quarter of the reserved roasted garlic on the center of each wrap. Divide the chicken, reserved beet, and roasted vegetables among the four wraps. Sprinkle each with a quarter of the cheese. Roll each wrap tightly in waxed or parchment paper, and slice in half before serving.