How to Make It
Stir together 12 cups water, apple cider, salt, soy sauce, brown sugar, garlic, kombu, thyme, rosemary, and bay leaf in a Dutch oven or large saucepan. Bring to a simmer over medium-high, stirring constantly, until salt and brown sugar dissolve, about 1 minute. Remove from heat, and let brine cool completely, about 2 hours.
Remove giblets from chicken, and reserve for another use, if desired. Place chicken in Dutch oven with brine. Cover and chill 8 hours or overnight.
Preheat oven to 300°F. Remove chicken from brine; pat dry with paper towels. Discard brine. Season chicken all over, including inside cavity, with 1 tablespoon salt. Stuff cavity with thyme sprigs, rosemary sprigs, lemon halves, and 2 tablespoons butter. Tie legs together with kitchen twine, and tuck wing tips under; transfer to a 12-inch cast-iron skillet. Grease one side of 2 large sheets of aluminum foil with butter, and coat with cooking spray. Place foil, butter side down, over chicken, and crimp edges on skillet to seal. Bake in preheated oven until a meat thermometer inserted in thickest part of thigh registers 150°F, about 1 hour and 50 minutes. Remove chicken from oven. Turn on broiler and allow oven to preheat 5 minutes. Remove foil, and return chicken to oven. Broil until skin is dark golden brown, about 12 minutes.
Remove chicken from skillet, and place on a cutting board; let rest 10 minutes (temperature inside chicken will continue to rise to 165°F). Place skillet on stovetop over medium-high. Add shallot and remaining 2 tablespoons butter to skillet, and cook, stirring often, until butter begins to brown and shallot is tender, about 3 minutes. Remove from heat, and stir in parsley, chopped thyme, lemon juice, pepper, and remaining 3/4 teaspoon salt. Taste and adjust seasoning, if desired. Carve chicken, and serve with skillet sauce and uncooked lemon halves.