Cover time and spectral gap for reversible random walks
Roughly speaking, the mixing time is the worst-case hitting time of the half the vertices.  Cover time is a stopping time when ALL subsets of vertices are hit.  In other words, it is always larger than the the mixing time.  Thus is your example one cannot have mixing time $n^{1000}$ and cover time $n^2$.  

Making this intuition precise requires a bit of care since we need to relate mixing times to eigenvalue gap, take not half the vertices but half the stationary distribution $\pi$, etc.  None of this is difficult.  Start with this paper by Lovasz and Winkler, which gives the above version of the mixing time and relates it to more standard mixing time in total variation.