I’m familiar (as a layperson can be) with metamorphic limestone in the form of marble, but it occurred to me the other day that there must be metamorphic rocks that originated as clays - and that maybe there are cases where these have something like the appearance or properties of fired pottery (albeit not in pot shapes).
Do such things exist? How common are they?
You start with clay and the sequence is shale –> slate –> schist –> gneiss
Seeing as clay is the most abundant sedimentary rock, they are all pretty common
Thanks - those weren’t quite what I was imagining though - not all clay deposits are conspicuously laminar, are they?
Clays found in one environment may exhibit little division, while those from another may readily form distinct bands. The laminae help move their classification into that of shales and are often accompanies by impurities such as iron, carbon, silica, limestone and other varied constituents. There is sometimes a tendency for lamination to occur with metamorphism and possibly, even in a relatively homogeneous sediment, the separate laminae may have more distinct mineral compositions, possibly reflecting those impurities in the suite.
For the record, my history is more with sedimentology and diagenesis. Hopefully, a proper petrologist will wander along shortly.
What are the chances of that? 
I’m not sure what you mean by conspicuously laminar. As Lieu says, some clays aren’t homogeneous; they can have differnt coloured bands, sandy layers etc. If by laminar you mean laminated (ie splittable like slate and to a lesser extent shale) then you are partially right.
Most clays are derived in large part from the weathering of older igneous rocks. One common ingredient in igneous rocks is the flaky mineral mica. This is hard to break to chemically and is present in tiny amounts in most clay. When the clay is buried, the tiny plates of mica tend to line themselves up horizontally, giving shale and slate its characteristic flakiness.
Thanks - What I was specifically looking for is clays that have been effectively fired to something resembling earthenware - does any such thing exist?
Round about here, a lot of the bedrock is argillite, which Wikipedia says is “a fine-grained sedimentary rock composed predominantly of indurated clay particles.” Does it look like earthenware? Well, in a manner of speaking. It’s kind of slatey.
…though it doesn’t split like slate. It tends to be of a uniform dark-gray color, though sometimes different, lighter-color layers are visible. I can’t find a good picture, though I think the foundation of my house is made of the stuff.
What you want would be very difficult for Mother Nature to produce. To make earthenware, you heat up clay to a very high temperature in a dry oven for a very short time. That’s a combination you just don’t see naturally. The processes bake the rock are going to happen underground, under pressure, There are going to be lots of mineral rich fluids floating around and everthing is going to take place over thousands of years.
That ain’t gonna make a coffee cup 
The closest thing you’re likely to find is some sort of hornfels, but it’s not that close.