An expert from Lunch Magazine: Just opposite the Church (Hallgrimskirkja) is a fantastic local cafe, perfect for sampling local cuisine. 

Cafe Loki is owned by Hronn, a former textile designer, who is passionate about Icelandic cuisine. Her husband Thorolfur bakes the famous rye and flatbread, a dense, sweeter version than the German, with a recipe that requires eight hours of baking at low temperature. Leftovers are used for bread soup and rye bread ice cream. Yes, rye bread ice cream. Hronn tells me ice cream is an Icelandic tradition, and most families have their own, unique recipes.

Cafe Loki's rye bread version is excellent, with bits of rye bread mixed with a creamy vanilla, topped with a dollop of whipped cream and rhubarb syrup. Other authentic treats include Skyr, similar to Greek yogurt (forget what's served in hotels and try the real thing here), and dishes like Icelandic Braveheart, schnapps, bread, dried fish, butter, and fermented shark. Easy to see how it got its name. Bites here won't break the bank it's perfect for a snack or casual meal.