The answer is Gregs Hut yes . Don,t know about lamb hill(if it is open). Gregs Hut can be noisy and crowded. Usually several Medics , John Bamber and a rescue team.. Everyone arriving at the hut tends to blow off steam with the relief of having survived Cross Fell., So not a very calm place to rest. The good news is that it is seriously warm as John has made it a tradition to lug a large bag of coal to the hut for use during the event.
Glowing stove ,hot noodles and good company make it a difficult place to leave.