It can be parsed that way in English, but essentially no one would form and use that sentence. The only way it would work is if the "breadbasket" in some way did something special to bread aside from containing it. So it would be possible say that the store is called "Breadbasket" and it serves bread from itself, but then its name would just be Panera.

To use that construction with its actual name being "Panera Bread" you would need to call it Panera Bread Bread, or Breadbasket Bread Bread.

To me it seems pretty clear that they wanted to name themselves Panera after changing their name from Au Bon Pain, and realized that they might have marketing issues as most people would not immediately know what either of those names meant. So they just tacked bread on the end of their new name to make it obvious what their specialty was.