Whisk 1 tablespoon olive oil, cumin, and coriander together in a large bowl; place quail in bowl and toss to coat quail completely.
Place 1 cured lemon slice in the cavity of each quail. Tie a piece of twine around the wings and under breast above the legs of each quail; tie legs together with twine.
Heat remaining olive oil in a large skillet over high heat. Cook quail in hot oil until browned on all sides, 2 to 4 minutes per side. Remove from heat. Turn each quail, breast-side facing up. Place 1 slice cured lemon on each quail breast and sprinkle each with cayenne pepper.
Roast in the preheated oven until slightly pink at the bone, 10 to 15 minutes.
Remove quail to a plate. Pour chicken broth into skillet and bring to a boil while scraping the browned bits of food off of the bottom of the skillet with a wooden spoon. Cook until reduced by half, about 5 minutes. Stir minced lemon into broth mixture. Return quail to skillet and pour sauce over quail until heated through, 2 to 3 minutes.
