I'm pretty sure something like that has been done at least once, somewhere. It probably doesn't happen super often due to the complications around IP ownership and conflict-of-interest regulations and what-not.
That said, if I were going to pursue something like this, I think the initial opening would be based on a personal relationship with somebody in the company. Preferably somebody as high up the org chart as possible. All I would do is go to $PERSON and say "Hey, can we get together for a coffee sometime soon, I have an idea I want to run by you." Totally informal, casual at this stage. If they meet with you, just lay out the idea in broad brush strokes "Hey, I have this idea I'd like to work on, it doesn't really fit what we do here, but I am interested in maybe standing up a company to build it, in partnership with $CURRENT_COMPANY. Do you think there's a way to make that happen?"
And just let it go from there. Whether or not this is a good idea and whether or not you should even bother hinges a lot (nearly totally, really) on how much you trust the people in your company and how much you believe they are reasonable people who generally want to do what's best for everybody. I've worked for companies where I had so little trust in upper management that I would never have bothered to approach anybody on something like this, and I've worked for companies where I could call the owner directly and run something like that by them. So really, lots of variation and "it depends".