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I don't really know what you consider "unusual" (other than fresh/store-made rather than packaged); I can't think offhand of styles of shortbread that would be available in Edinburgh but not elsewhere in the UK. However, millionaire's shortbread (shortbread layered with caramel and chocolate) is yummy, and this place appears to have it (and still be open, unlike other Google hits). From what I can see they appear to do their own baking. The recommendation was from 2013, though, so you might want to phone or facebook-message them and check before going out of your way ...