A ‘piece’ is generally a sandwich, regardless of filling. What the English might know as a ‘chip butty’ is known in Scotland as a ‘chip piece’ for example.
It’s said that in the mid 20th century, children living in tenements, especially in Glasgow areas such as Easterhouse or Drumchapel, would shout up to their ‘maw’ (mother) for a snack, whilst they played outside, which was then launched out the window for the children to catch. The ‘piece’ most often making its way out the tenement windows was a ‘jeely piece’ (jam sandwich), a fact immortalised in the ‘Jeely Piece’ song, composed by the late, great Matt McGinn, which contains the excellent chorus about the often wayward nature of the ‘pieces’:
‘Oh ye cannae fling pieces oot a twenty story flat,
Seven hundred hungry weans will testify to that.
If it’s butter, cheese or jeely, if the breid is plain or pan,
The odds against it reaching earth are ninety-nine tae wan.’
Similarly, a ‘playpiece’ was what children took to school for break, and even as late as the mid 1990s, distraught children could be heard in school plyagrounds across the central belt lamenting their forgotten ‘playpiece.’