What would you do? Freeze is on, but top performer's demanding a raise

Periodically, we ask three HR managers how they’d handle a difficult situation at work. Today’s problem: The company’s top performer says she’s leaving unless she gets a raise — but there’s a wage freeze on.
The scene
HR director Doug Webster was just about to leave for the day when supervisor Pete Mangione appeared in his office door.
“Got a minute?” asked Pete.
“Uh … sure, Pete,” said Doug. “What’s up?”
“Maureen Mills is putting a gun to my head – that’s what’s up.”
‘Not exactly a team player’
“Never figured Maureen for the firearm type,” Doug retorted. “A club would be more likely. What’s she up to now?”
“She’s threatening to quit if we don’t give her more money. I’ve explained we’re under a freeze. She says she doesn’t care, she’s our biggest producer and she deserves more money. And if she doesn’t get it, she’s going to take her services elsewhere.”
“That’s our Maureen,” said Doug. “She’s a heck of a salesperson, but not exactly a team player.”
“That’s for sure,” said Pete. “But she’s sort of got us between a rock and a hard place. She is by far our leading performer – I’d hate to think where we’d be if she left.”
“And if she did decide to leave, she’d certainly have no trouble landing another job,” said Doug. “She’s got a non-compete, so she probably can’t hurt us that way. But we might lose some of the accounts she’s handled for years.”
“Great. If we don’t knuckle under, she goes, and we lose business. If we do cave, we’ll have a revolt on our hands,” said Pete. “We can’t give in to her, can we?”
If you were Doug, what  would you say or do next?
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What your peers said
Jerry, HR director, Great Lakes, IL
What Jerry would do: We wouldn’t cave to Maureen’s requests, but we would look into other, non-monetary benefits we could offer, such as additional vacation time or a possible cash award at the end of the year.
Reason: There’s a wage freeze on, and we can’t go back on that. Maureen may be the top producer, but she’s not the only producer. We need to assume our product will stand on its own, with or without her.
Will, HR director, Kingstreet, SC
What Will would do: I’d do a comparative pay analysis on all employees in Maureen’s job position. Looking at compensation and length of tenure with the company, I’d see where she stood compared to other workers with similar responsibilities. If she’s making less, we might be able to justify a pay raise. But if she’s making more, we could tell her she’s already doing better than similar workers here, and we can’t give her more money.
Reason: Maureen is asking for money, so I’d home in on exactly what she’s asking about. If that fails, I might branch out into alternatives, but right now, money is the issue at hand.
Vivian, director of HR, Aurora, CA
What Vivian would do: I’d get together with Pete and Maureen to talk about her point of view and figure out what we can do to help. At the same time, we can’t just give her a raise, so maybe a dialogue will help us find a reasonable solution.
Reason: My job in HR is to look at the big picture for the whole company. We recognize Maureen’s a great producer and we’d hate to lose her, but we have to be fair to the other workers as well, so a compromise is our best shot at coming to an understanding.