I already have great working relationships with several staff engineers in my org, but I'd like to take it a step further and enter a formal mentorship with them. How can I go about this in a natural way?
Something I tell a lot of people is to treat your manager as a people API. Whenever it comes to improving/fixing relationships, your manager should be on point for that, and establishing a mentorship falls under "improving relationships" for sure.
Tactically, just bring it up in your next manager 1 on 1. You can say something like, "I've been working with staff engineer X, and I respect them a lot and we get along really well. I think I could really level up quickly if they became a formal mentor of mine. Can you reach out to them to see if they're interested in this, and if they are, set the expectation that I'll put a recurring 1 on 1 on their calendar soon?"
Some other things to mention in this conversation:
A staff engineer's time is extremely valuable, so your goal is to make it a slam dunk case for your manager that they should totally help set this up. This means painting a picture of why it's a good fit, and if you succeed in doing this, the incentives all line up. There is no manager who wouldn't want their senior engineer to blossom into another strong staff engineer on their team - That's more impact they can land via their reports.
The good thing about your scenario is that it seems like you have multiple mentor candidates in mind. If your top choice doesn't pan out, you can ask about the other ones.