Kouign-amann is a French pastry whose name literally translates to "cake, butter". The texture of this little cake is more that of a pastry than a cake. When you take a bite it feels like puff pastry with butter melting in your mouth along with a nice sweetness. The little cake dates back to 1860 and was conceived in the town Douarnenez located in Finistère, Brittany.
If you'd like to try your hand at making this dessert; here is a recipe. The cake that is shown on this page is from the Manresa Bread bakery in Los Gatos, California.
While I'm not sure there really is a substitute for a freshly baked Kouign-amann but there are some alternatives if you're not up to the task of baking your own and you don't have a great bakery nearby to rely upon.
Trader Joe's does [did] offer a suitable substitute but as of 2017 that product was discontinued. Alternately if you don't mind doing a little baking then you can visit Drizzled and try their cheater version of Kouign-amann which uses a frozen puff pastry, butter, palm sugar and honey.