Bring 4 quarts water to a simmer in a large stockpot over medium-high. Remove from heat; whisk in 1 1/2 cups salt and sugar until dissolved. Let cool completely, about 1 hour. Add chicken; cover and refrigerate at least 30 minutes or up to 8 hours or overnight. Remove chicken from brine; discard brine. Place chicken, skin side up, on a wire rack set inside a baking sheet; pat chicken dry. Refrigerate chicken, uncovered, 8 hours or overnight.
Preheat oven to 375°F. Trim 1/2 inch from top of each head of garlic to expose cloves. Place garlic on a large sheet of aluminum foil. Drizzle 1/2 teaspoon olive oil over each head; wrap tightly with foil. Place on a baking sheet; transfer to preheated oven, and roast until cloves are tender and light golden, about 45 minutes.
Squeeze cloves from 5 garlic heads into a food processor; discard skins. Reserve remaining roasted garlic head for chile-basil vinaigrette. Pulse garlic until finely chopped. With processor running, gradually drizzle in 1/3 cup olive oil, processing until smooth, about 1 minute. Add mustard and leaves from 2 thyme sprigs; pulse to combine. Rub garlic mixture all over chicken halves; refrigerate, skin side up, uncovered, at least 1 hour or up to 8 hours or overnight.
About 1 hour before serving, preheat oven to 400°F with racks in the upper and lower third positions. Place a large rimmed baking sheet on top rack while oven preheats. Toss together potatoes, onions, rosemary, 2 tablespoons oil, 2 teaspoons salt, and remaining 3 thyme sprigs in a large bowl.
Remove chicken from refrigerator; wipe off excess marinade. Heat 2 tablespoons oil in a large, heavy skillet over medium. Add chicken, skin side down; cook until deeply browned on bottom, 6 to 10 minutes. Flip chicken, and transfer skillet to lower rack in preheated oven. Roast until a thermometer inserted in thickest portion of meat registers 155°F, 30 to 40 minutes.
While chicken roasts, add potato mixture to preheated baking sheet, spreading in an even layer. Return to top oven rack; roast 20 minutes. Meanwhile, toss broccoli planks with crushed red pepper, remaining 2 tablespoons oil, and remaining 1 teaspoon salt. Remove pan with potatoes from oven; arrange broccoli on top of potato mixture, and return to oven. Roast until broccoli is crisp-tender and potatoes can be easily pierced with a fork, 10 to 15 minutes.
Remove chicken from oven; increase oven temperature to high broil. Broil broccoli mixture, watching carefully, until well charred, 3 to 5 minutes.
Transfer broccoli mixture to a large platter; drizzle with lemon juice. Place chicken on top of vegetables, and smother with chile-basil vinaigrette. Sprinkle with basil; serve with additional vinaigrette.
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