1. Pour lima beans into a colander; sort and remove debris, then rinse and drain beans. In a 4- to 5-quart pan over high heat, bring 2 1/2 to 3 quarts water and beans to a boil; cook for 2 minutes. Cover tightly, remove from heat, and let stand for 1 hour. Drain beans, reserving 2 quarts cooking liquid.
2. Rinse chicken and pat dry. Cut each thigh in half lengthwise along one side of bone; or with a cleaver, cut whole chicken through bones into 2- by 3-inch pieces (see notes). Pull off and discard any lumps of fat. Rinse green beans and remove and discard stem ends and any tough strings; cut beans into 3-inch lengths.
3. If using a charcoal barbecue (at least 22 in. wide), ignite 160 briquets (about 9 1/2 lb.) on the firegrate; open dampers. When coals are dotted with ash, in 15 to 20 minutes, spread into an even double layer about 2 inches wider than base of paella pan. Set grill 4 to 6 inches above charcoal. Let coals burn down until they're hot (you can hold your hand at grill level only 2 to 3 seconds), 5 to 10 minutes.
If using a gas barbecue (at least 19 in. wide), turn temperature to high, close lid, and let heat for 10 minutes.
4. Set a 17-inch paella pan on grill over hot coals or high heat on a gas grill. Add 2 tablespoons oil; with a wide spatula, spread to coat bottom of pan. When oil is hot, add chicken pieces (close lid on gas barbecue); cook, turning once, until lightly browned on both sides, 6 to 12 minutes total.
5. Add tomatoes and stir until most of the juices have evaporated, 1 to 2 minutes. Stir in remaining 1 tablespoon oil, green beans, 1 1/2 teaspoons salt, paprika, and turmeric. Add soaked lima beans and the reserved 2 quarts cooking liquid (if it's cold, heat in a microwave oven or pan until hot before adding). Note level of liquid in pan: Measure distance from pan rim or handle rivets to level of liquid. (Close lid on gas barbecue.) Bring to a gentle boil.
6. Boil gently to flavor broth, 25 to 30 minutes (on gas barbecue, regulate heat as needed to maintain gentle boil). Remove about 1 cup broth from pan. Add enough water to match original level of liquid, 3 to 5 cups. Taste broth in pan, and season to taste with more salt; also taste reserved broth and season with salt.
7. Pour rice in a band down center of pan. Draw a wooden spoon or spatula in a wide zigzag pattern through rice and across pan to distribute rice evenly. Push any grains floating above broth back under. Simmer, without stirring, until rice is tender to bite, 25 to 30 minutes. If rice begins to scorch before it's tender, remove pan from heat and let coals burn down to a lower temperature before continuing. (On gas barbecue, close lid and regulate heat so liquid simmers briskly in the beginning, then lower heat when most of the liquid has evaporated; rotate pan occasionally so bottom cooks evenly.) If rice is still firm, drizzle about 1/2 cup of the reserved broth evenly over the top and cook for a few more minutes; repeat if necessary. (If reserved broth isn't needed, save for another use.)
8. With spoon, push aside a little of the top layer of rice to see if there is a brown crust on the bottom; if one hasn't formed yet, cook a few more minutes (adjust gas heat as necessary to brown, but not burn, rice). Remove from heat and let stand for about 5 minutes. Squeeze juice from lemons over rice. Add salt to taste.