What do you do when a high-performing employee, whom you can’t afford to lose, wants to quit for remuneration reasons?

I don't know about the difference of what the employee gets now and what others are willing to pay. If the differences aren't that great, it's rarely just a matter of salary. From my experience, it is often the culture of the company or department. I am permanently confronted with the fact (public funds) that employees can earn significantly better elsewhere. Nevertheless, the fluctuation is very low because I offer a variety of opportunities. 
No silos, free choice of projects, maximum freedom to make own decisions.

Even if you can convince the employee with money not to terminate the contract, you have to assume that he has quit in his head. The step of announcement has already happened. Better you try to eliminate the reasons why you can not do without him - as soon as possible

First of all, I would try to assess the request against known / existing remunerations within the domain and other companies.
Second, remuneration quite often is a major but not the only reason. So the company's culture and current ways of treating employees should be questioned especially when high-performers are about to quit.

Need to analyze his or her current role, how that role is more aligned with organizational goal and strategy, if his or her role value add to the organization goal and the long term vision/strategy of then i will really think of consider and offer a counter offer. But it will not be necessary in all roles and designations. In some cases a person is given a counter offer based on how important the skill set for the project and the market demand for such skills. In all the cases the reconsideration only works well if he or she is considering the only on the basis of financial gain.