EDIT: In this question I forgot to put one of the assumptions, and the question was easier than it should be. Here is the revised question. Please vote to close this question as it is no longer relevant.
Question. Let $G$ be a finitely generated non-amenable discrete group, and $H$ be a subgroup of $G$ of infinite index. Can it happen that the index of the normalizer $N(H)$ of $H$ in $G$ is finite greater than $1$, and the Schreier graph of $G/H$ has subexponential growth?
If the answer is yes, I would very much like to see an example. It would be especially nice if $G$ could be taken to be a property $(T)$ group.