Is there a strongly connected graph where 2 vertices exist such that random walk from one to another requires more than polynomial time?

Question

The question is fairly simple: find an example of strongly connected (directed) graph $G$ on n vertices, such that degree of every vertice is at least $\frac{n}{2}$ and there exists a pair of vertices $(s, t)$ such that random walk from $s$ to $t$ takes more than polynomial time. I don't have any clues to solving this problem, so any help will be great!