Create a component template of what that role is supposed to see, create a component on the server of that, grant browse access to the component but not to the server.
Ahh, I thought I had given Component.* to all the components. I had given full access to the Template but not the components. It works now! Thanks Erik for your help!
The trick is that you need to give the role Server.read access to the server (so that the ACLs will contain entries for users in that role), and Component.Browse access to the Component (so they can browse the custom parts of the server you specify).