Here are my thoughts. The color of the cog reminds me of a bulk film loader for 35mm film. There is indeed a grey cog inside one brand that I found a picture of, but it is not right. Of course, the TIE cog was painted, but that could have been so that they all matched one specific one.
Any local camera store might have current and vintage film loaders, so you can look at various ones. The design has not changed since they started making the plastic ones, though most brands probably aren't around anymore. Every high school and college photo lab had DOZENS of these, going back decades. And we know the props people on ANH used lots of photography-related found parts. Some of the brands of film loaders are Watson, Prinz, Arista, Alden, Western, and many others you can find with a search for Bulk Film Loader.
The center notches on the TIE cog hub indicate that it accepted a crank handle, EXACTLY like the bulk film loader removable handle. Although on the film loader shown here, I think the crack goes to a different part, but it would fit exactly to the cog center hub notches, it seems.
Other items could have accepted a similar crank handle, such as the Visible V8, Visible Wankel, or some other model with gears and cogs.
It could also be from a toy with gears, cogs, and crank handles, similar to these I have shown here for inspiration. I realize they are not the right cogs in these toys, it is just to spark ideas for everyone. Though it seems more likely to be from a model kit than from a toy.
Happy hunting!
