What it means
A piece is a sandwich, usually one you’ve packed for work or school. You’ll hear piece and jam for a jam sandwich, and play piece for the one kids scoff at interval. It’s everyday Scots, dead normal, and if someone asks what’s for your piece, they mean the filling, not a slice.
Usage examples
"Dinna leave without your piece, you’ll be starving at playtime. I’ve made it cheese and pickle, and there’s a wee juice in your bag."
"Mam packed me a tuna piece and a tangerine for the school trip to Edinburgh Castle, the lid of the box flew off in the wind on the esplanade, and the tuna ended up in a seagull beak before I even tasted my own sandwich properly."
"Granny made the best pieces in Drumchapel, soft white bread, real butter and a slice of corned beef from the deli down the road, and the wee box of them lasted exactly until the playground bell at quarter past one every day."
"You taking a piece the morra or you buying something from the shop at lunch"
"He opened his school bag and everything smelled of egg piece for the rest of the bus ride"
Where it comes from
This is proper Scots and it goes way back. Piece originally meant a bit or portion of food, especially bread, and in Scotland that sense settled into everyday use for a sandwich you take with you. It never sounds fancy or old-timey there. It’s just normal lunchbox language.