Pottery man, have taken a lot from your detailed reply which all makes perfect sense.
Tried to collate as much info as possible, so here goes.
Piece pictured is 9mm thick, 60 mm long and 20mm wide, normally on an Id request do use the gridded measurement scale.
As said 2 parallel grooves aprox 5mm apart and about 5mm from top of this piece.
The remnant is virtually flat, slight arch of 1mm max on the inside
No water around, no known Roman habitation, slight slope.
Did find a few other pieces in an area of 70m x 30 m.

The above is the smoother sides,

Top piece10mm, 2nd piece is just to support, 3 is 9mm, 4 is 12mm bottom is13mm.Did find 4 Roman coins in this area 300-350 AD.
Hope this is of more use to you.
Cheers,
Dave.