Tapping on a chef will open their kitchen and a panel at the bottom of your screen with all the available dishes to cook.
Drag a dish across to the kitchen to start cooking. You will need to make sure you have produced all the ingredients before you can start cooking or you can purchase the missing ingredients with Gems.
Items showing a hand with a serving tray in the panel indicate that a table requires this dish.
When your dishes are ready they will appear in the Chef's kitchen.
When the storage is full your Chefs will continue to cook some dishes, however once the quota is reached the production in the kitchen will be on hold until you free up some storage space and tap on the Chef to collect the dish that is ready.