Without seeing a picture, I can only guess, but since you were at a deli, I suspect you were offered what's called a 'deli roll'. They're a type of oval roll, often split in half to make sandwiches. Did it look something like this?
If so, it was a 'deli roll'.
The regular square/rectangular piece of bread would be called a 'slice' or 'piece' of bread.
Also note, that in your original post, you can't say "came with a bread' - bread is non-count. The omelette/omelet came with bread.
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