Here's how to get golden eggs in Forager: feed any chicken with the Animal Feed item. This is produced by using 2x Flower + 2x Jelly + 2x Poop at the Mill. Create this and feed it to a chicken, and it will have a chance of giving you a golden egg instead of a regular one.
The drop rate for golden eggs is still pretty low, so you might need to create a lot of Animal Feed items before you get it, since these are consumables. So remember to not sell all that Poop you eventually find in your game, and save it to produce Animal Feed instead. It can be a bit of a hassle to get the golden eggs in Forager, but just keep making Animal Feed, and you will eventually get lucky.
It's also a small chance for the Chickens to drop Golden Eggs without the use of Animal Feed, but this is extremely rare.