There are so many places that this is just a personal, extremely non-professional list of those places where you should sit down for a decent cup of coffee, enjoy the view and the vibe of the not-so-small-town-Dalmatia.
1. Academia - one of the staples of social life in Korčula, a bar with live music in the evening during the summer season is actually at its best during the morning hours (that means sometime before 2 pm). Many locals and tourists alike will not be able to resist sitting under the trees where the bar's terrace is located, either on their tables, or on makeshift furniture, having breakfast with their coffee while seeing the sea, the boats (small and large) docking, it's the centre of all that's happening in Korčula during the day.
2. Galerija - this is another one of those places that is mostly full, and those having coffee there are people who don't care that much about being seen, rather enjoy good coffee. Galerija doesn't really have a sea view, just a nice cosy temperature that makes you feel good, decent coffee and it's not overly popular with the tourists, so it's possible to blend in with the locals for a while.
3. Servantes - another bar that turns into live-music place late in the evening (they organize festivals as well, and have a wonderful little stage set-up in the park), but is also extremely popular for a morning enjoyment, because of the lovely shade provided by the trees in the park, a lot of movement in the street just opposite the bus stop where it's located, and people recounting the events of the last evening, which might make you feel like you were there although you went to sleep at 10 pm.
4. Hotel la Ville - this is a more classical solution, the terrace of the oldest hotel in Korčula, a four-star beauty with an almost perfect location on the werstern side of the Korčula Old Town, this is the place to have your morning coffee if you want to be seen by the important people in Korčula. You'll meet the mayor here, or any celebrities that are in Korčula at any point. But, please, just make sure you don't sit on the first table right of the entrance, as that is the table where for decades the distinguished gentleman of Korčula, a very exclusive group, have been meeting to solve all the world's problems. They haven't yet succeeded, obviously, but with their determination, it's just a matter of time!
5. Kavana No1 - a less traditional choice, but also the bar in Korčula with one of the most amazing views, if you're into the view of the entire entrance to the channel, all the boats around the islets and ferries coming and going, the bar at the first floor of the Tommy shopping centre might be the place for you. Supposedly, the coffee there is the best in Korčula (but as you already know, that is something that changes very rapidly, so even if it were true last year, there's no guarantee that it will stay that way) and the view is breathtaking. And you'll be away from the rush of a busy tourist town, while basically still in it.
