Is $\frac{n}{\log n} \log \frac{n}{\log n} = O(n)$?
Clearly, $\log(\frac{n}{\log(n)}) \le \log(n)$.
Substitute this into the equation to get that $T=O(n)+\frac{n}{\log(n)}\left(\log\left(\frac{n}{\log(n)}\right)\right)\le O(n)+\frac{n}{\log(n)} \log(n)=O(n)+n=O(n)$
So $T=O(n)$, and indeed it is not bigger than linear time.