On day 1 (2 days before Thanksgiving), remove giblets from turkey and rinse turkey inside and out. Set turkey, breast side down, in a large disposable roasting pan set on a rimless baking sheet. Add giblets to pan (discard neck). Into cavity of turkey, sprinkle coarsely chopped ginger and garlic. In a bowl, whisk together minced garlic, minced ginger, soy sauce, vinegar, and brown sugar; pour over turkey. Cover turkey with plastic wrap and marinate 2 days, basting 3 to 5 times a day.
On day 3 (Thanksgiving Day), remove giblets and 1 cup marinade from pan and set aside for Portuguese Sausage Dressing. About 4 hours before serving, remove turkey from refrigerator. Let stand for 30 minutes in pan; meanwhile, soak hickory chips in water 30 minutes.
Remove turkey from pan and pour marinade into a bowl; add chicken broth and set aside. Return turkey to pan, breast side up, and add onions (put 2 or 3 quarters inside turkey). Truss turkey, tying drumsticks together tightly with kitchen twine; truss wingtips together the same way. Cover wingtips and drumsticks with heavy-duty foil "caps," molding them snugly to prevent scorching. Cover entire pan loosely with a double layer of foil, extending foil beyond turkey like an umbrella.
Prepare a charcoal or gas grill, large enough to hold turkey, for indirect cooking: On charcoal grill, ignite about 60 charcoal briquets. When coals are spotted with ash (about 20 minutes), mound equally on opposite sides of firegrate. To each mound, add 5 unlit briquets and 1/2 cup drained soaked wood chips now and every 30 minutes during smoking. If using gas grill (with at least 11 in. between indirect-heat burners), put 1/2 cup drained soaked wood chips in a drip pan directly on heat in a front corner and add 1/2 cup chips through grate every 30 minutes during smoking. Turn heat to high and adjust gas for indirect cooking. Close lid and preheat grill 10 minutes.
Carefully slide pan with turkey from baking sheet onto center of grill. Pour in marinade and broth. Close grill lid. (If your grill has a lip on it, rest pan at an angle on grill, then carefully slide pan onto grill.) Smoke turkey, basting every 1/2 hour, until an instant-read thermometer inserted straight down through thickest part of breast to bone registers 150°, about 2 1/2 hours (begin checking after 2 hours). Ladle marinade from pan into a heatproof bowl and reserve for Hawaiian-Portuguese Turkey Gravy and Portuguese Sausage Dressing.
Slide baking sheet under turkey pan, ease baking pan onto sheet, and lift pan with turkey off grill. Let turkey rest, covered with foil, at least 20 minutes (internal temperature will rise to 160°) before carving.
Note: Nutritional analysis is per serving.