Technically I don’t think this is a ‘kitchen’. That term almost always describes a room inside a house. Outhouse, not so much either.
An out building I guess is a good enough description.
In the south meals were cooked in another building probably until the late 1800’s. The large homes in Charleston also had back buildings. I can’t imagine having fires all summer long. Too hot, besides everything burned down here at some point.

These photographs are of a back building preserved at a plantation.

I’ve never seen the doors open here so it may be storage now. The fact that it exists at all is nice.
Santee Delta, South Carolina.