If you see this a * next to the dish when a customer orders or on the order board this means that they would like the upgraded dish to the amount of stars required * or ** or  ***.

You will not be able to serve them a dish you have ready in your storage without the star.

However if you have upgraded a dish you can serve this to a customer who wanted the dish with no stars.

To be able to upgrade your dishes you will need to collect ingredient essences. 

You collect these from the wholesale delivery which you can get for free by playing the game or if you can not wait you can use Gems to buy them.

Once you have collected  enough to upgrade a dish a ! will show on top of your Academy.

Here it will show you a list of chefs you have. Next to the chef if you can upgrade a dish that they cook it will have a green arrow.