These are from his Smoke and Mirrors collection, which I’ve been reading. They’re slight, and to my mind the shorter one works and the longer one doesn’t. But they illustrate a storytelling device that for some reason hadn’t quite made it into my head.

The device is this: a story will feel more like a story, a complete narrative, if its ending features some element that was also featured by its beginning. The echo or chime makes the reader feel like something complete has been told.
Humorists do something like that too. At least Calvin Trillin did: his last paragraph would always bring back some joke featured earlier in the column. Stand-up comedians pull the same trick. So it’s not like the full-circle device is a new discovery — we’re just talking about closure, right? But I don’t read a lot of stories, and for me it was an experience to catch the device at work. 
The better of the two stories, “Daughter of Owls,” was just a couple of pages and was written in the style of John Aubrey, which helped; not that I claim to have read Aubrey, but the style was fun and novel enough to give the piece some float, and then the full-circle device came along and buckled everything into being a story. Without the device, we would have this: way back in olden times, a mysterious girl is left by owls in a small town, and when she gets old enough the men of the town rape and kill her, and then the owls come and kill the men. With the device we have this: the owls who left the girl also left typical owl dung consisting of pellets that held small animals’ bones and skin, and then the owls who killed the men left dung pellets that contained the men’s bones. Story! (Pretty much; Gaiman’s telling helped a lot.)
The longer story, “Shoggoth’s Old Peculiar,” is buckled this way: an American boy having a dismal time hiking in Britain is fed up with his travel guide, which keeps promising decent inns, friendly people, etc.; he stumbles into an especially unpleasant village, where he has a mysterious encounter with the supernatural; he wakes up, the village is gone, no one has heard of it, and the relevant page has somehow gone missing from his guidebook; back home in America, he writes a letter to the guidebook author, not only to give her a piece of his mind about the book but also to ask about the mysterious town; he’s relieved when he never hears back.
The buckle is the guidebook, which is entirely ancillary to the story’s action. Nothing happens because of the book, its author doesn’t play any role in events, but hauling the book back in again still works well enough when it comes to holding the story together. The chime is still enough.
I find it Gaimanesque that mentions of the book should all be funny until the story’s very end, when the book has a final mention that’s played for quiet unease. To turn one feeling into another, amusement into fear, for example, is the sort of device Gaiman likes to work (or the  sort of whatever — I mean that it’s something he does). I think it adds to the, oh dear, silvery quality his work can have, the sense that various tiny givens of reality are too uneasy to stay put and that reality is always shifting at the corner of your vision. Why that adds up to “silvery” is another question, of course.
So, to recap, the buckle in “Shoggoth’s Old Peculiar” works even though the item used to make the circle isn’t at the heart of the story’s action. So the problem with the story is elsewhere. Basically, “Shoggoth’s” is meant to be Pete and Dud meet Cthulhu. That is, Gaiman imagines Peter Cook and Dudley Moore doing one of their addled-old-duffers crosstalk dialogues but sitting in an English counterpart to Innsmouth and using Lovecraft’s stuff as their subject. That’s all right as an idea, but what he comes up with isn’t much, just a color-by-numbers pastiche of Pete and Dud and a couple of commonplace observations about Lovecraft (yeah, he used big words). 
Also, I didn’t go for the story’s opening about what a bad time the fellow was having in England and how he hated his guidebook. High-spirited, comic Gaiman doesn’t add up to funny Gaiman, in my experience.