Turing-unrecognizable language - what TM does?
You might be a little confused by a definition somewhere.  A TM $M$ is a recognizer for a language $L$ if it 


Accepts every string in $L$
Either rejects or loops on every string not in $L$


Now let's think about the language ${\overline{A_{TM}}}$.  An input $\langle N,x\rangle$ is in ${\overline{A_{TM}}}$ if either:


$N$ rejects on $x$.
$N$ loops on $x$


So any recognizer will need to figure out (in finite time) that $N$ is going to loop on some input, which is just the halting problem.