me too! That was stress....the little mutt I could live without ( but our son would be sad, it's his dog) but Otto would be very bad. He cost a small fortune, and guards the house and is MY dog. I just lost my previous one last year ( she was old but I was still super upset) but with her I knew it was coming, she just got old and died but this one is still almost a puppy.
I've had this issue a few times with my sheppards and one other dog. Pretty sure your dog jumped a deer. Rabbits run in circles when chased by dogs, other small game doesn't go far, certainly not out of ear shot. A white tail deer... will run to water, a river or swamp. They will run several miles, well out of voice range. And they run in a straight line, as terrain allows. It only takes 3 or 4 minutes for a dog to be out of hearing range. Your dogs couldn't hear you call.
For me... deer run for the swamp via 2 routes. I know where the roads are that cross those routes. When I realized mine jumped a deer I'd try to get ahead and be waiting.
The problem with sheppards... they will chase silently. Hounds will bark as they run, easy to figure out where they are headed. With sheppards you need to know where the deer will go when chased.
Solution... ask hunters in your area which way deer run when chased... they'll always follow the trails and routes they normally travel. Once you know where these routes cross roads then you'll know how to get ahead of your dogs and cut them off.
Some dogs get a taste for chasing deer, hard to break them... every dog is different so I wish you luck. I've had dogs be gone 3 days, come home cut up from briars, thorns. A pain to deal with them.
The dog in this photo... was a real problem. I never managed to completely break him from chasing deer so... I bought a radio tracking collar so I could find him. I've used it on another dog, she was smart though, quickly learned not to chase deer. If you can catch them in the act of chasing a deer then you can break them. But only while they are doing it. Punishing them after the fact, after they've returned home doesn't work.
Edit to add... take a good look at a topo map of your area. Deer tend to run down hill when chased (but not always, they will go over a ridge to get to their destination)