Born in Manchester, Leigh to Rushmoor school in Bedford before going to Clifton College and Christ's College, Cambridge in 1968. After Cambridge he tried his hand at farming and novel writing before falling into cooking "almost by accident" in 1977.
After a couple of years at the Joe Allen restaurant, Leigh went to work with the Roux brothers at Le Gavroche in 1979. After stints at Le Gavroche and the brothers' pastry laboratory, and becoming buyer for the group, he took over their prestigious Le Poulbot restaurant as head chef in 1984, receiving many accolades including The Times "restaurant of the year" award in 1986.
He opened Kensington Place restaurant with Nick Smallwood and Simon Slater in 1987. Quickly hailed by The Times as restaurant of the year, Kensington Place and its blend of brilliant food and an informal and buzzy atmosphere set the pattern for London restaurants in the 1990s.
Leigh remains cookery correspondent of the Financial Times. His much accoladed book, No Place Like Home, was published in 2001. He left Kensington Place in December 2006 in order to open Le Café Anglais in 2007.