It was called "Big Bird's Birthday or Let Me Eat Cake", and aired on March 15, 1991.TVTropes.org wrote:Cookie Monster resists to eat Big Bird's birthday cake...by eating everything else on Sesame Street. Not all the food. Everything. At one point, he was shown actively eating a lamp post. The show cuts back from one of the skits, and the camera panned over Sesame Street, which looked more like a war zone.
Part 1: https://www.youtube.com/watch?v=-cj1a2ByU-M
Part 2: https://www.youtube.com/watch?v=kegnwmCzcBo
Part 3: https://www.youtube.com/watch?v=Zu5FCifWPp4
Part 4: https://www.youtube.com/watch?v=Z84YcgZ5S28
Part 5: https://www.youtube.com/watch?v=oMixkuuzEBY
Part 6: https://www.youtube.com/watch?v=l29FhCxEnis
And as a bonus... https://www.youtube.com/watch?v=nrr89vJyyws