Your alien surgeons remind me of the Thaed (death spelled backwards!) in E. C. Tubb's "Talk Not at All", originally published in Nebula Science Fiction, No. 33, 1958 (U.S. edition dated December, 1958), which is available at the Internet Archive, reprinted in the anthology Gryphon Science Fiction and Fantasy Reader #1 (Philip Harbottle, ed.).
Quoting the editorial blurb from that 1958 Nebula:
The Thaed were the engineers of life, working upon
living tissue as the Earthly metallurgist fashions steel
From the story:
"The Thaed are masters of life," I said dully. "They are biological engineers working with living materials as we work with ores and minerals. As surgeons they are, and have ever been, unequalled anywhere in the known universe. They are an old race, how old even they have forgotten, and it may well be that they were responsible for adapting and spreading life as we know it." I hesitated, not liking what I next had to say. "It would even be reasonable to give them the credit for our creation."
I paused, wondering just how I had acquired this knowledge. Not for the first time I wondered just what they had built into my shattered body when I lay helpless in their chambers. A certain amount of conditioning, that I now knew, but what else?
"They are an old race and a decadent one. They have retained their manipulative skill but it has turned upon itself. They distort flesh and bone to create objects which have no real place in the scheme of things and they do this for no other reason than that it interests them to do it."
And that was true. I had reason to know it. Words such as "amuse" could not be applied to the Thaed any more than words like "help" and "hate". But how could anyone explain the Thaed? How can anyone explain what it is that makes a man watch a line of marching ants and then, for no reason, either step on them or remove an obstacle from their path?
"Anyone can come to the Thaed," I continued. "And the Thaed will give them what they demand. But there is one stipulation. The Thaed will either give you what you want in the way you want it, or they will give it to you in a manner decided by themselves. In either case they will keep to the letter of your demands." The air seemed to have grown thick around me. "Those terms apply to any and everyone who lands on this planet."
[. . . .]
"Seven years ago I landed here in a wrecked ship with two companions." I forced myself to speak calmly, hard though it was. "We were injured, all of us, injured in the worst possible way. The pile had blown and what wasn't broken was rotting with radiation sores. Anywhere else we would have died within hours of landing."
"So we wanted to live and the Thaed gave us what we wanted. I was the lucky one, I lived the way I wanted to, in the shape and form of a man. The others—well, they lived, too, after a fashion. If you can call it living."
"The tapestry!" Clitheria shuddered as she stared at it. "That thing—?"
"Jud Owens," I said bleakly. "Once a man and now what you see. They showed him to me before I left."
"And the other one?"
"I don't know. I don't want to know. The only thing I am certain of is that he is alive and conscious and that every second of every day he prays for death."
Unfortunately, the story details you provide do not match "Talk Not at All." Maybe Tubb wrote another story about the Thaed. If so, it might be hard to find, because he wrote a lot of stories.