I have a version of this on the Appalachian Folks' self-titled album (NOT ON LABEL, 1976) which I suppose is ultimately closest to what Fiddler's Companion calls MOLLY PUT THE KETTLE ON [4]. The tune is similar to the Yew Piney Mountains tune in the beginning, though vocalist Byard Ray sings a verse I haven't found elsewhere. After playing the tune three times over (and singing once to it), the band instrumentation drops to just rhythm and he sings a close (though still unique) verse more in line with what you'd expect from Polly Put the Kettle On to a slightly different five verse rhythm. You can hear a live recording of it here, though he doesn't sing any of the verses.

This (click) YouTube video by user jadefiddle (fiddler Jason Cade, presumably) explains that Ray got the tune from Manco Sneed. He elaborates further in the digital liner notes to his album Hog-Eyed Man Vol. 3 (link):

Byard Ray (1910-1988) of Madison Co., NC, got the gist of this one from Manco Sneed; another that Manco had learned from J.D. Harris. Over the years, Byard’s version diverged into a few variants, which we arranged into a single setting for our recording on volume 3. Jason's mother took lessons with Byard Ray when Jason was young, and Byard even put a rattlesnake rattle in Jason' fiddle to give it more mojo.

My transcription of the lyrics is as follows, preceeded by the brief entry from the liner notes:

We think this may be one of the oldest tunes we play. It shows all the lonesome sounds of the banjo and fiddle when played in a modal tuning.

Well I catch my horse, I'll go riding
Make some meal when the sun goes down
I'll get my Polly a rule-eyed devil
And we'll get married in the nearest town

Well, you ought've been around when Polly put the kettle on
Polly put the kettle on with a smile
Well, you ought've been around when Polly put the kettle on
To smell the aroma I had to buy