There are three important keys in Crow Country, each one based on the, er, cardinal precious metals; bronze, silver, and gold. The gold one is the trickiest to find, but one of the easiest to get. Maybe. Let me explain.
The gold key requires two things to find: access to the theatre and either a flamethrower or a (full) bottle of acid.
The theatre’s main doors are locked from the other side when you first find them. You need to wait until you have access to Haunted Hilltops, at which point you can go through the northeast staff hallway (reached through the Manor), and enter from the back. Then, you can unlock the door.
The key itself is in the head of the fairy sitting on the edge of the stage. If you shoot her in the head, it will shatter revealing the gold key encased in resin.
To get the key out of the resin, you have the two options I mentioned: acid and the flamethrower. You can get those two items around the same time, but in my opinion, the flamethrower is the easiest to obtain.
You find the flamethrower by completing the puzzle in the Train Room, but that requires that you have the Data Disk. Once you do, you have to solve a puzzle where you identify the train cars on the completed train. It’s easily my least favorite puzzle in the game. You can get more details on it in my guide for completing all the secret puzzles, but to save you time, the code from the top to the bottom is BDAC. Ugh.
Once you have the flamethrower you just point it at the resin blob and hose it down with fire. It’ll melt and you can grab the key.
Your other option is to get the acid. You find the bottle for the acid in the Dungeon. It also requires the Data Disk for you to obtain it. Once you have it, you also need access to the underground. Once there, you can go to the Roots room marked “Friday.” There’s a spigot there that you can use to fill up the bottle. Once you have it, go back to the theatre and dump it in the sprinkler tank in the corner. Turn the valve and it will spray on the fairy’s head and melt it. As I said, the flamethrower is a bit easier to get.