Adjust 1 oven rack to lowest position and second rack 8 inches from broiler element. Heat oven to 250°F. Line baking sheet with aluminum foil and place wire rack on top. Sprinkle chicken pieces with 2 teaspoons salt and season with pepper (do not pat chicken dry).
Heat oil in 12-inch skillet over medium-high heat until shimmering. Place leg quarters skin side down in skillet; cook, turning once, until golden brown on both sides, 5 to 7 minutes total. Transfer to prepared sheet, arranging legs along 1 long side of sheet. Pour off fat from skillet. Place breasts skin side down in skillet; cook, turning once, until golden brown on both sides, 4 to 6 minutes total. Transfer to sheet with legs. Discard fat; do not clean skillet. Place sheet on lower rack, orienting so legs are at back of oven. Roast until breasts register 160°F and legs register 175°F, 1 hour 25 minutes to 1 3/4 hours. Let chicken rest on sheet for 10 minutes.
While chicken roasts, sprinkle gelatin over broth in bowl and let sit until gelatin softens, about 5 minutes. Whisk water and cornstarch together in small bowl; set aside.
Melt butter in now-empty skillet over medium-low heat. Add shallots and garlic; cook until golden brown and crispy, 6 to 9 minutes. Stir in coriander and cook for 30 seconds. Stir in gelatin mixture, scraping up browned bits. Bring to simmer over high heat and cook until reduced to 1 1/2 cups, 5 to 7 minutes. Whisk cornstarch mixture to recombine. Whisk into sauce and simmer until thickened, about 1 minute. Off heat, stir in parsley and lemon juice; season with salt and pepper to taste. Cover to keep warm.
Heat broiler. Transfer sheet to upper rack and broil chicken until skin is well browned and crisp, 3 to 6 minutes. Serve, passing sauce separately.