Giving more protein won't stop them tearing up the lawn. They may hear grubs or something just under the surface and they are after them. They will hunt down live protein if they can. If you don't want them to tear up the lawn, you are going to have to confine them to their own area.
My 3 Rhode Island Reds have their own garden area with a beautiful stand of Blue Grass and it short order all the grass was gone so now there is nothing there but the large Rose of Sharon they like to hide under.