I don't know about BBQing pigeons but I am familiar with the idea of BBQing geese in flight. Many, many years ago I was working on a USAF project using the high powered 1,000,000 watt radar at Hanscome Field near Boston. That was the first one that bounced radar signals off the moon The technician in charge of the radar had the fantasy of catching a flight of wild geese in the radar beam and cooking them before they hit the ground. One day he managed to catch a goose in the beam about a quarter of a mile away walking across the runway. He flipped the switch and blasted the goose. Te goose shrugged and walked away. Apparently no one ever told the technician about the inverse square law. The goose just got a mild hotfoot. It would have to be sitting on top of the antenna before it cooked.