Carrots are usually sliced. We might refer to the slices of carrot as pieces of carrot or slices of carrot. If a child called that "carrots" I might correct him/her, and I might not not. (I don't know.) Potatoes are sliced into small pieces for putting into a stew. (When you make stew everything is mixed together, so it seems a bit odd to ask for one thing.) We usually eat several peas at a time, so we wouldn't ask for one pea.
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