Central Grocery (not "Market") is definitely the home of the muffuletta. One of the secrets is the Sicilian bread for which the sandwich is named. In NOLA, the standard is made by Leidenheimer Baking Co. If you can't get theirs, make your own (recipes available on line).
If you're in New Orleans, remember that Central Grocery is closed on Sunday and Monday. I think their regular hours are 9-5.
There will be an excellent opportunity to meet up at Central Grocery for a no-host muffuletta lunch on Thursday, April 24th. SWMBO and I will be in New Orleans for JazzFest (starting the next day) and maybe some of the rest of you can plan to join us. More about that later and in a separate post.