Preheat the oven to 500°. In a saucepan, combine the soy sauce, mirin, sake, vinegar, brown sugar and ginger. Add the cornstarch slurry and bring to a boil over high heat. Cook, stirring, until glossy and slightly thickened, 3 minutes. Transfer the teriyaki sauce to a bowl.
Set the turkey in a large roasting pan; scatter the shallots around it. Season the turkey cavities and skin with salt and pepper. In a small bowl, blend 4 tablespoons of the butter with the olive oil and brush some over the turkey.
Roast the turkey for 30 minutes, or until golden. Baste with the butter mixture and add the water to the roasting pan. Reduce the oven temperature to 325° and roast the turkey for 1 hour, basting twice with the remaining butter mixture; loosely cover the bird with foil if the breast browns too quickly.
Pour half of the teriyaki sauce into a bowl; baste the turkey with some of it. Roast the turkey for 1 1/2 hours longer, basting with the sauce from the bowl every 30 minutes; the turkey is done when the skin is lacquered and an instant-read thermometer inserted in an inner thigh registers 170°. Transfer to a carving board; let rest for 45 minutes.
Meanwhile, strain the pan juices into a bowl, skim off the fat and reserve the shallots. Set the pan over 1 burner. Add the shallots to the pan and cook over high heat, stirring, until browned, about 3 minutes. Add the pan juices, Rich Turkey Stock and the reserved teriyaki sauce. Bring to a boil, scraping up any browned bits from the bottom of the pan.
Strain the pan sauce into a medium saucepan, reserving the shallots. Boil the sauce over high heat until reduced by a third, 30 minutes. In a bowl, mix the remaining 2 tablespoons of butter with the flour until smooth. Whisk the flour paste into the sauce and boil, whisking constantly, until the gravy is thickened, about 5 minutes. Add the shallots, season with salt and pepper and transfer to a warmed gravy boat. Carve the turkey and serve with the shallot gravy.
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