Is refuting candidate deciders of the halting problem computable? No Turing machine can decide whether any given Turing machine will halt for a given input. That is:


If you give me a Turing machine which you claim can take a Turing machine and an input for that machine and will tell me whether the machine will halt,
There is some counterexample TM/input pair I can provide to that machine for which it will fail.


This is the undecidability of the halting problem. Old news.

My question is: Can such a counterexample be computed for any candidate decider of the halting problem?

That is, is there a Turing machine which can take as input some Turing machine and compute a TM/input pair for which the given Turing machine fails to decide the halting problem?

(Incidentally, I'm slightly over my head here, so I may be asking my question wrong, or asking a wrong question. Thoughts and pointers are welcome.)
The usual proof that the halting problem is undecidable already gives you exactly such an algorithm. Given an algorithm $A$, we construct an algorithm $B$ that on input $x$ computes $A$ on program $x$ and input $x$, and then enters an infinite loop iff $A$ answered "halts". Now consider giving $B$ itself as an input. If $A(B,B)=\text{"halts"}$ then $B$ doesn't halt on $B$, so $A$ is wrong. Same if $A(B,B)=\text{"doesn't halt"}$. So $\langle B,B\rangle$ is an example input on which $A$ is wrong.