1. Heat a grill, ideally charcoal, to medium (about 400°). Meanwhile, bring broth, 1/2 tsp. chile flakes, and the chicken to a simmer in a 5- to 6-qt. covered pot. Reduce heat and simmer until chicken is barely cooked through at bone, 12 to 15 minutes.
2. Transfer chicken to a bowl (save broth for other uses). Let stand a few minutes, then drain juices. Coat chicken with oil, remaining 1 tsp. chile flakes, and the salt. Coat lemons on cut sides with the oil in bowl.
3. Grill lemons on cut sides until softened and grill chicken, turning once, until browned, with grill covered, 5 to 10 minutes total; if chicken flares up, move to another spot. Transfer to a platter. Garnish chicken with thyme.
Make ahead: Up to 1 day through step 2, covered and chilled.