When I was in Copenhagen, we went to a restaurant, Koeford, that specialises in ingredients and dishes from Bornholm, and it was a really lovely evening. Our waiter, Lars, was from there too, and looked after us well, bringing a series of dishes to the table that though they were basically traditional, were served with contemporary style (perhaps the presence of Noma, "the world's best restaurant" just across the harbour made them up their game). So we had things like our entree of herring and quail's egg served in a little preserving jar with elm smoke trapped in with it, that escaped when we opened the lids. Some dishes were served on small island-shaped slabs of Bornhom granite: rooster, forest pork, mushrooms, leeks, really nice bread with soft buttermilk butter, celery crisps... Courses between courses. Yum.
Even the water on the table came from the island; same with the beer and wine matched to the dishes, but best of all to my taste was the Bornholm cider (made, Lars informed us, by a co-operative of disabled people, which was nice but also rather ironic, given the effect that cider has) that was served with the dessert of calvados-flavoured rhubarb sorbet and tangy oxidised apple. Double yum. And then chocolate to finish, with a sticky wine. Glorious! And fortunately, the restaurant was within staggering distance of our hotel; on balance (actually, the lack of it), it was as well we hadn't gone to Noma, which we'd been sad about at first. We ate just as well, and there was no risk of falling overboard on the way back. Which is always a good thing.
UPDATE: And, in the grand tradition of TravelSkite coincidences, what should I get in the mail the morning after posting this, but the winter catalogue from Ecco shoes with this photo at the front, with a blurb that says, "...and our favourite place... is the cozy, charming island of Bornholm - a small, mountainous haven in the Baltic Sea..." Amazing.