Burgess began working for Cooper, both in their factory and at their drivers' school based at Brands Hatch and raced one of the works Formula Two cars in 1957, when he achieved fourth place in the Oulton Park Gold Cup. This led to a drive with Tommy Atkins' team in 1958, with a similar machine. He won at Crystal Palace and Snetterton and gained fourth places at Reims and Montlhéry. However, a broken leg at AVUS ended his season but not before he made his Formula One debut for Cooper at the British Grand Prix.