Add 4 cups water, Adobo, brown sugar and bay leaves to medium pot over medium heat. Cook, stirring, until Adobo and sugar dissolve. Transfer Adobo mixture to container large enough to fit turkey and all of liquid. Stir in Naranja Agria and remaining 16 cups very cold water. Submerge turkey in brine, breast side-down. Cover and refrigerate for at least 8 hours and up to 16 hours. 2 hours before cooking, remove turkey from brine. Discard brine and rinse turkey inside and out with cold water; pat dry with paper towels and let come to room temperature.
Heat oven to 450°F. Place turkey on rack in large roasting pan; pat dry once more with paper towels. Stuff orange quarters, onion quarters and oregano sprigs into turkey cavity. Tie legs together with kitchen twine, tuck wings behind back and tuck excess skin from neck into top cavity. Using pastry brush, brush turkey skin with olive oil. Pour 2 cups chicken bouillon mixture into bottom of roasting pan. Roast turkey 30 minutes; baste turkey with pan juices and rotate pan. Reduce oven heat to 350°F. Roast turkey, basting with pan juices every 30 minutes and rotating pan for even cooking as necessary, until skin is dark golden brown and crisp and turkey is cooked through (internal temperature registers 165°F when inserted into thickest part of thigh without touching bone), about 3- 3½ hours, adding remaining chicken bouillon mixture in 1 cup measurements if pan becomes dry and tenting with foil if turkey is browning too quickly. Carefully remove rack with turkey from pan; loosely tent turkey with foil to keep warm before carving (at least 30 minutes and up to 1 hour). Pour pan drippings into large measuring cup; remove and discard fat. Reserve 4 cups total remaining pan juices and remaining chicken bouillon mixture for gravy.
While turkey cooks, heat 2 tbsp. olive oil in large skillet over medium-high heat. Add onions. Cook, stirring occasionally, until coated in oil. Reduce heat to lowest setting. Cook, stirring occasionally, until onions soften and deeply caramelize, about 30-45 minutes. Stir in flour; set aside in skillet until turkey is cooked.
Reheat onion mixture in skillet over medium heat and cook until flour starts to brown, about 2 minutes. Stir in balsamic vinegar, mixing well. Slowly add reserved pan juice mixture, stirring constantly to prevent lumps. Bring onion mixture to boil; cook, stirring occasionally, until gravy thickens, about 5-7 minutes. Transfer gravy to serving boat. Or, for smooth gravy, blend until smooth then transfer to serving boat.
Carve turkey and serve with gravy.