The name of her current chef is not publicly known - the names of royal servants often aren't well known until they leave their jobs - but until recently Queen Elizabeth's chef was Darren Mc Grady. He also created some special dishes for her Jubilee in 2012, and has recently given some interviews about what it was like cooking in Buckingham Palace.
It seems that she isn't a great foodie, so cooking for her wasn't that difficult. She likes quite ordinary food like cereal or boiled eggs for breakfast, fresh fish, simple vegetables and so on. Her favourite thing apparently is very small jam sandwiches known as jam pennies.
However, he obviously had to cook some very grand meals for official occasions, and has created banquets for top politicians, heads of state etc. Darren McGrady now lives in the USA and still works as a chef.
You can try some of his recipes here.