Does there exist an undecidable problem such that the answer is YES for exactly one input to a UTM, and NO for all others? Suppose I have a universal Turing Machine (UTM) which accepts some input in binary. Is there a computational problem such that the answer to the problem is YES (accepting) for exactly one input (and hence NO for all others), but determining which input is the accepting one is undecidable?
Or is there a fundamental reason this cannot happen? Or is determining if this  problem exists undecidable?

I assume this is equivalent to asking: is there a property of TMs such that this property is true for exactly one TM and not true for all others such that determining which TM this is true for is undecidable?
For every Turing machine over some language L, there is a set of words that it will halt and accept on and a set of words it will halt and reject on, if it does halt.

If $|L| = 1 $ then the Turing machine will accept on exactly one input.

Given an arbitrary M and some input, we cannot decide whether the program will halt; halting problem. However, since M is not arbitrary, it’s fixed,this is not the Halting problem. I believe an algorithm would only need to check if the input is equal to the first element in L.