Bring 2 quarts cider, 1 cup salt, and the next 10 ingredients to a boil in a very large (16-qt.) pot, stirring to dissolve salt and sugar. Let cool to room temperature. Stir in 1 1/2 gallons cold water. Add chicken to brine and press down to submerge. Cover; refrigerate overnight.
Remove chicken from brine and pat dry with paper towels; discard brine. Season lightly inside and out with salt and pepper (Chef’s note – since a brine contains a lot of salt, if you are salt sensitive, use salt sparingly.) Place chicken, breast side up, on a rack set in a large heavy roasting pan, stuff with oranges and tie legs together with kitchen twine. Let stand at room temperature for 1 hour.
Preheat oven to 375°. Combine the remaining 1 cup of cider and 3 cups water in roasting pan. Brush chicken with butter. Flip breast side down.
Roast chicken, breast side down, basting occasionally, for 40 minutes Using paper towels, flip chicken. Roast, basting occasionally, for another 30-45 minutes, until an instant-read thermometer inserted into thickest part of thigh registers 165°. Transfer chicken to a platter. Let rest for at least 20 minutes before carving.
Meanwhile, strain the juices from the roasting pan into a saucepan. Simmer over medium heat until juices have thickened, about 10 minutes. Serve the cider jus alongside the chicken and garnish with cinnamon sticks.
