At my last startup, one of our employees approached this well. This is a very summarized version, but he came to me and said: "I'd like to be paid $X. I think I deserve it and I believe I can get that salary on the open market. I don't want to negotiate on this."
It was a very significant raise but after a short discussion with my cofounders, we agreed and gave him what he asked for. That he was a top performer made it an easy decision.
Doesn’t tend to work at big tech as they have rigid very slow moving (and ill informed) HR processes that only react after they see a bunch of people leaving.
In this particular case, I imagine that person would have left and found another job. They were certainly talented enough to do so. If so, there would've been no hard feelings on my end given that they're just going for what they deserve. If the reason was only because we wanted to but couldn't afford to, then I hope there wouldn't have been any hard feelings on their end either since we had a positive relationship and I would've made it clear to him that we wanted to retain him but simply didn't have the funds to.
It was a very significant raise but after a short discussion with my cofounders, we agreed and gave him what he asked for. That he was a top performer made it an easy decision.