Examples for directed graphs with super polynomial cover time

Question

The cover time of a graph is the expected number of steps in a random walk on the graph until we visit all the nodes.

For undirected graphs the cover time is upperbounded by $O(n^3)$. What about directed graphs? I'm looking for examples of super-polynomial cover time.

Is there an example for such graph with $O(2^{\sqrt n})$ cover time?