So during the Korean Folktale Topic Challenge, I was actually searching for a specific story that I thought was associated with a mythical beast called "Haetchi" or a "Haetae". It's been a while, and I may be confusing the story with the story of the "Bulgae". The basic premise of the story is a dog trying to find its owner after he doesn't come home (because the owner died, I think). The dog tried to ask the sun or something for help? Something happens and it ends up chasing the sun, but it never does. I think it was a way of Korea explaining the day/night cycles. Anybody have any idea of the story? I didn't have any luck finding anything.
The closest match I can find in Myths and Legends from Korea: An Annotated Compendium of Ancient and Modern Materials by James H. Grayson (Routledge, 2001) is Tale 100: "The Origin of Eclipses" (pages 253-255). This story describes a Dark Land neither sunlight nor moonlight. Dogs were raised in that land that were known as "Fire Dogs" (Grayson's translation does not use the term Bulgae).
The king of the Dark Land sent out the strongest Fire Dog to steal the sun. However, the sun was too hot, so the dog had to drop it and return to the Dark Land to explain the problem to the king. The king then sent out the dog to steal the moon, but the moon was cold as ice so the dog had to spit it out. The Dark Land is still submerged in darkness. When the Fire Dog tries to steal the sun, people [elsewhere] call this an eclipse of the sun; when the Fire Dog tries to steal the moon, people call this an eclipse of the moon.
The book contains a total of 191 myths and legends, some of which involve dogs, but none comes closer to the description in the question.