There are 2 different kinds of Chef Hats: Silver and Gold. You get Chef Hats when you complete a Catering Service (starting from level 30) or when you complete a Pool Party (starting from level 45).
In addition, you can win Chef Hats at the Fortune Hut. You can also buy Chef Hats with Cash. You need Chef Hats to try your hand at the Fortune Hut and activate Speedy Service in your restaurant. Apart from this, you can also buy some decors using Chef Hats and train few Staff members in Academy.