"My instinct was to take the reservation: He had the most elegant voice I had ever heard," says Peire, 56. "But I was new, and the general manager was standing in front of me shaking his head."
Peire offered the caller a table in the bar, with a promise that he would do everything possible to find space in the dining room on the night.
"This is Sir Alec Guinness," said the Academy Award winning actor, best known as Jedi Master Obi-Wan Kenobi in Star Wars. "I have never taken dinner in a bar in my life and I don't intend to break that rule tomorrow. I shall dine elsewhere."
So much attention is devoted to chefs nowadays, it is easy to forget that in a fashionable restaurant, the maître d' generally decides whether you get a table, at what time, and where you sit. In many ways, he is more important than the chef, a fact that used to be widely recognized.
"To have a famous maître d’hôtel greet you respectfully by your surname, to greet him in turn familiarly is a strong tonic for your ego,” Vogue said in 1936. The magazine is quoted on the website of Quaglino's, which itself was founded by a maître d', Giovanni Quaglino.
A good maître d' will make an instant assessment of how much effort to make to accommodate you when tables are as prized as tickets for the Centre Court at Wimbledon.
Just ask three of the leaders at London dining rooms: Peire of the Ivy, Jesus Adorno of Le Caprice and Silvano Giraldin of Le Gavroche. Elena Salvoni of L'Etoile, the best-known woman maître d', died this week after a short illness. She was 95.
"The maître d' is like the host of a party," says Gibraltar-born Peire, who is a director of the Ivy and known for his TV appearances as The Restaurant Inspector. "You have got to decide on the evening who feels, looks and sounds like they are going to fit in with the party. If they don't, and you can afford to turn them away, you turn them away.
"People always think there are secrets about how to get a good table, but no. You need to look pleased to be there. You need to give a good impression. You listen to what people have to say. You ask the right questions. You dress appropriately. Don't overdress. Be friendly. It always helps if you talk to staff. There are regular customers at the Ivy only because they are charming to the staff.
"Most of our regulars are not famous people. We just treat them all like stars. Another thing: You need to go back to places and have a list of, say, five restaurants you support regularly. And when you go back to those restaurants, you'll be treated like a friend."
He loves his job and reckons he can assess within seconds which table someone will be joining from the guest's body language, the way he dresses and the way he talks.
"A maître d' can make or break a restaurant," he says. "I do see, every so often, some of my colleagues in restaurants once they make it to the maitre d' status, they think they are god's gift to customers.
"But if you are snooty or have a bad attitude with the customers, the ones who are paying your wages, your mortgage, you will never really make it in a restaurant with longevity," says Adorno, 62.
Across Mayfair, Silvano Giraldin, 67, was born in Padua, Italy, worked at Le Gavroche for 37 years and remains a director. For decades, he was the gatekeeper of the restaurant, where his favorite guests included Charlie Chaplin and Princess Diana.
The princess used to dine with Richard Attenborough at table 15, but never returned after paparazzi gathered one night, and she suspected Le Gavroche had tipped them off - an idea that Giraldin dismisses.
Even celebrities can't always get what they want.
The Rolling Stones were regulars in the early days of Le Gavroche until the restaurant introduced a strict dress code. They didn't return. Barry Humphries was untroubled when warned in advance that gentlemen must wear a jacket and tie. He simply turned up in a voluminous frock, dressed as Dame Edna Everage.
Was he allowed in?
"Of course," Giraldin says. "You can't turn away a lady."
Back at the Ivy, Peire is keen to emphasize that his focus is on service not celebrity.
Fair enough, Fernando, but just one more anecdote please.
"Someone rang and said, 'This is Michael White. Do you have a table in 10 minutes?' I didn't recognize the name - he was a producer - but the way he asked meant either that he knew his name carried weight or that he had no idea where he was calling.
"The Ivy was like Rick's Café in Casablanca in those days. Everybody wanted to come.
"I took the risk that he might be our sort of customer, which he was because he arrived with David Bowie and his wife. They looked extraordinary.
"David was wearing a beautiful pink suit. I remember talking to him about how beautiful his shoes were. They were big and pointy and he'd had them hand-made in Switzerland. And Iman had had her hair teased for a photo shoot. It was enormous, like a massive light bulb on her head.
"They were utterly charming and the pair of them cut a real dash.
"They made everybody's night."
Richard Vines is chief food critic at Bloomberg. Follow him on Twitter @richardvines