When to prune, I'm not sure. I just got some beeches up in this place and I haven't read anything about them.
What I do know, is that usually, beech hedges in the Netherlands are trimmed in the end of spring (when leafs have developed) or in the start of autumn. They usually bounce back pretty well from that. We don't have hot summers like you do, and our climate is a bit wetter, I guess.
I would try to get closer to the trunk, so chop nearly every branch in half. This helps in making secondary and tertiary branching closer to the tree. Eventually, that would help fill it up. I personally feel like the branches are a bit too long now. It will fill up out of proportion (small but wide tree with a lot of empty space).
But again, I don't know anything about beeches yet. Mine haven't gotten out of dormancy, so I'm not doing any work on them until the autumn.