The comic "Kessel Run" initially saw publication in Star Wars Tales 16 and was subsequently included in the compilation Star Wars Tales Volume 4.
Squishmael, a Sloogarian who trades anecdotes for beverages, fondly recalls the evening Han Solo triumphed over Lando Calrissian to claim ownership of the Millennium Falcon. Han is under the impression that he has "made it big", but Lando retorts that he must establish a solid reputation if he desires to ascend to the ranks of elite smugglers, emphasizing that being a top-tier smuggler requires more "than a fast ship". Lando proceeds to propose the Kessel Run as a means of enhancing his standing, and Han confesses (to some amusement) that he is unfamiliar with it. Lando and his companions clarify that Kessel is not a system, but rather a species of avian creature, and that completing the Kessel Run entails capturing enough of these Kessel to fill a cargo hold and then delivering them. The central question becomes: how far will he need to travel? Will he succeed in "fifteenth parsecs...or thirty?" Given that the existing record stands at approximately 13, Han accepts the challenge and sets off in the direction of the system. He receives communication from a mysterious individual offering assistance with his mission, claiming to have associates who are currently looking to offload Kessel.
Han consents to compensate for the information and agrees to pay a modest sum of 1000 credits for the coordinates of the initial planet. He agrees to this as a test to ensure that everything appears to be "legitimate." Upon arriving at the designated planet, he discovers the contact awaiting him with the container of Kessel Birds prepared for transport. Han is then persuaded to part with an additional 3000 credits to acquire the first shipment, based on the assurance that they are worth "no less than 12000." He complies and resumes his journey. After yet another payment for the second location and, once again, an escalating payment for the birds, he tallies up 17000 credits expended and reaches the final destination at 11.8 parsecs. He approaches the bartender in search of the buyer he is meant to meet, only to discover that the bartender is the very individual he was sent to find. Initially relieved, Han proudly declares, "Well, Dahk, break open the history books. You're looking at the man who just made the Kessel Run in less than twelve parsecs!" Dahk announces this to the patrons of the bar, and everyone erupts in laughter, as it turns out that Han has merely become "another Kessel Runner," signifying that this group routinely defrauds individuals of their funds.
Han and Chewie execute a hasty departure upon realizing that they have, in fact, acquired a worthless bird and squandered a considerable sum for nothing more than a few laughs. Meanwhile, on the opposite side of the galaxy, Lando and his cohort divide the spoils and share a hearty laugh over the day's events. Lando does experience a pang of guilt for extracting so much money from his friend, but Squishmael rationalizes it by asserting that Han was allegedly cheating during the game in which he won the Milleneum Falcon. Squishmael then attempts to continue his storytelling, inquiring whether the reader wishes to hear a tale of how he and Darth Vader were disrupting some Tauntaun on Hoth. This concluding remark suggests to the reader that the entire account was likely a fabricated fabrication from a storyteller attempting to wheedle his way into a few complimentary drinks.
For this narrative to align with established continuity, it must transpire before Han Solo has any prior knowledge of the Kessel Run, and after he has secured the Millennium Falcon from Lando Calrissian. According to The Hutt Gambit, Solo first executed the Kessel Run in 5 BBY, yet he did not win the ship from Calrissian until 2 BBY. Consequently, this narrative appears incompatible with canon, except perhaps as an in-universe embellished story recounted by Squishmael.