This song and that rasp were made for each other.
Waits is singing here with the Preservation Hall Jazz Band, and the tune is "Tootie Ma Is A Big Fine Thing." It first appeared on the early 2010 compilation Preservation, a benefit album for Preservation Hall featuring tons of special guests (Del McCoury, Andrew Bird, Ani DiFranco, Dr. John, Merle Haggard, Steve Earle, etc.). It's also available, as you can see, on a limited edition 78 rpm record along with the B-side "Corrine Died On The Battlefield." That disc comes out this Friday, Nov. 19 (in New Orleans) and Saturday, Nov. 20 (online). Those not in Louisiana can visit the band website for more info.
Wait, a 78 rpm single? You mean those heavy suckers from before 1950? Yup. If you don't have a turntable capable of playing 78s, you can buy one of 100 custom record players from Preservation Hall's deluxe bundle. All I'm saying is that you New Orleaneans better not shut me and my AT-LP120 out on Friday.
Both songs are old Mardi Gras Indian tunes. For source recordings, see the Baby Dodds album Jazz A La Creole, referenced a number of times on HBO's Treme. (It's Danny Barker's group who plays them, but Dodds' name is on the jacket.) You might also know "Tootie Ma" from the Medeski Martin & Wood version with John Scofield.