I think you will be hiking along the Sierra Nevada, not multiple Sierras and certainly not possessive Sierra's.
There are about three large agencies that you might have to deal with. One is the National Park Service (NPS) which is part of the Department of the Interior. They control the national parks and monuments, and they issue permits for wilderness use there. One is the National Forest Service (NFS) which is part of the Department of Agriculture. They control the national forests which often surround the national parks, and they issue permits for forest wilderness use there. Additionally, there is the Bureau of Land Management (BLM), and they issue permits for BLM land in some cases.
In some cases, a national forest may have some small special area included within it, like a named roadless area or wilderness area, so sometimes there are special rules and permits. However, the national forest permit station can issue any permit that is required. In some cases they have daily usage quotas. In a few spots, they have do-it-yourself permits.
Yes, it is supposed to be that you can be issued one single permit that authorizes you for a thousand miles or more. You may be delving into the depths of government bureaucracy to get that done, but that's the way it is supposed to work in theory.