Golden Egg #15 is located in Level 1-12: Using the green bird destroy the donut on the right side of the screen. The Golden Egg will then drop down and start swinging back and forth. With some luck and good aim use one of the remaining birds to hit the Golden Egg. I actually found it easier to hit it on it’s way back (swinging from left to right). To each his own though. (→ see image)
Note, this is a two part video. The first part shows how to find the Golden Egg and the second is a walkthrough on how to pass the actual Golden Egg level.