Want to replace faucet handles, but the broach is a mystery

Hello,

I want to replace the handles on my laundry sink with something bigger, such as a wrist blade. I didn't think that would be rocket science... until I found myself unable to identify the faucet brand or find anything resembling the shape of the broach. When I showed the handle to a hardware clerk, he said he'd never seen anything like it.

As you see in the pix, it's a generic-looking faucet with no identifying marks:

http://www.plumbingforums.com/forum/attachment.php?attachmentid=13708&stc=1&d=1481259930

The broach has three flat sides forming a rectangle, and a fourth rounded side:

http://www.plumbingforums.com/forum/attachment.php?attachmentid=13709&stc=1&d=1481259930
http://www.plumbingforums.com/forum/attachment.php?attachmentid=13710&stc=1&d=1481259930

Can anyone identify it, and suggest where I might find handles to fit?

Thanks!

One mystery solved: faucet is a Moen 74998, with a Moen 1224 cartridge. Why there appears to be only one handle in the world to fit that cartridge is one I may not solve, and may have to just replace the faucet.

Delta, Moen or Valley

This is a diagrams of a Moen