Position an oven rack in the bottom third of the oven. Preheat the oven to 400°F.
Rinse the turkey pieces and pat them thoroughly dry with paper towels. Brush all sides with olive oil, then season evenly with kosher salt and black pepper.
Arrange the turkey pieces skin-side up on a baking sheet. Place the thick ends of the breasts and thighs toward the corners and the drumsticks in the center.
Roast for 1 hour and 10 minutes.
Remove the turkey from the oven and generously brush it with the cranberry orange glazed turkey mixture. Pour the chicken broth onto the baking sheet around the turkey to help prevent the drippings from burning.
Return the turkey to the oven and roast for another 20 minutes. If the skin becomes too dark, loosely cover the turkey with foil.
Check the thickest part of the thigh with an instant-read thermometer, making sure the thermometer does not touch the bone. Cook until it reaches 165°F to 170°F.
Transfer the turkey to a serving platter. Brush generously with additional glaze and let it rest for 25 to 30 minutes.
While the turkey rests, scrape the pan juices and browned bits from the baking sheet into a large measuring cup. Carefully spoon off and discard the excess fat from the surface.
Transfer the degreased pan juices to a medium heavy saucepan and heat over medium-high heat.
Whisk in 2 to 3 tablespoons of the cranberry-orange glaze, or more to taste. Bring the mixture to a gentle simmer and cook briefly until the flavors come together.
Slice the turkey and arrange it on the platter. Garnish with fresh herbs, orange wedges, and whole cranberries. Serve warm with the homemade gravy.