Adjust oven rack to lower-middle position and preheat oven to 275°F. Spread onions, carrots, celery, garlic, bay leaves, and black pepper evenly across bottom rimmed baking sheet. Pour 1 cup broth baking sheet (reserve remaining broth). Place rack on top of baking sheet (it may rest directly on top of vegetables—this is ok). Set aside.
Rub turkey breasts, legs, and wings with 2 tablespoons vegetable oil. Season liberally with salt and pepper. Place on top of rack on top of vegetables, trying to leave a little space between each turkey piece.
Roast turkey until instant-read thermometer inserted into center of breast piece registers 150°F, 2 to 3 hours total. Remove breast and set aside. Continue roasting legs until instant-read thermometer registers 170°F, about 30 minutes longer (depending on arrangement, legs might finish at same time as breasts). Turkey pieces will not be browned. Set aside at room temperature for at least 30 minutes and up to 2 hours. Set fine meshed strainer over large bowl. Strain vegetables from tray. Reserve liquid and discardvegetables.
While turkey is roasting, roughly chop carcass and neck into 1-inch chunks. Heat remaining 2 tablespoons vegetable oil in large saucepan or Dutch oven over high-heat until just beginning to smoke. Add turkey carcass and cook, stirring frequently, until well browned on all surfaces, about 10 minutes. Deglaze with remaining stock. Add tap water until turkey pieces are barely covered. Bring to a boil, reduce to a simmer, and allow to simmer while turkey roasts, skimming off scum occasionally.
While turkey is resting, pour stock through same strainer and add to reserved stock from strained vegetables. Discard solids. If desired, cut giblets into 1/4-inch pieces. Heat butter in medium saucepan over medium high heat until melted. Add giblets and cook, stirring frequently, until lightly browned, about 2 minutes. Add flour and cook, stirring constantly, until flour is light blond in color. Slowly whisk in reserved stock. Bring to a boil and reduce to a simmer. Cook until desired consistency is reached, about 10 minutes. Season to taste with salt and pepper.
30 minutes before serving turkey: Preheat oven to 500°F. Place turkey in oven and cook until crisp and browned, turning breast once during cooking. Serve immediately with gravy.