I see Peregrines fairly frequently. But they’re usually two long avenue blocks away on the tall church. It’s much rarer to see one perched in Green-Wood.
The metal top of St Michael’s RC church at 4th Avenue is a long-time perch and butcher’s block. The other day, one Peregrine was up here eating (probably pigeon) and another was circling around. This one pictured here had just bathed and was grooming those deadly feathers.
Backyard and Beyond 
I believe falcons deliver the coup de grace with the specially notched bill designed for the purpose. The spine is severed from behind dispatching prey efficiently if not painlessly.