Go into the stage with respect and humility. Whatever level of experience you have, show great respect for everyone you meet, pay attention to what you are told and when corrected, keep your mouth shut and follow orders.
You can relax a bit by doing as Nicko suggested and simply doing your best. You will not be allowed to do anything they don't feel you are ready for. So you will start out with simple tasks and move on to more challenging tasks as they see fit.
What to look for.
Throughout your stage, however long it may last, watch carefully how the kitchen operates. Note the total number of people in the kitchen, note what their tasks are, how the work area is set up, everyone from dishes to expeditor.
How do they communicate with each other, how is instruction given.
If you happen to see a delivery, note the amounts of things, what kind of product it is. How often is the delivery. Is there any negotiation between the purveyor and the chef? Does the chef check the order for quality, inspect the items, return or refuse anything? Why did he/she return or refuse the item?
Note the behavior of the cooks. How neat they are, how focused they are. How do they interact with each other?
Note the behavior of the staff before, during and after service.
Note the interaction between cooks and waitstaff before, during and after service. How do they communicate? About what?
Get a copy of the menu. Note the number of items on it. See how the items are divided up among the cooks for production. How many menu items are the responsibility of each station?
Note when prep is done. How much prep for each item? Are the recipes complicated or simple? What gets labeled and in what way is it labeled?
How noisy is the kitchen? What is making the most noise? Is everyone talking? About what?
If you see something you don't recognize, when you have an opportunity to do so, ask what it is.
What tools, in addition to knives, are being used and for what? How does each menu item get plated? Is it a simple or complicated plating? How may cooks does each plating require?
Note simple techniques. Note the quality of each ingredient. How does the ingredient change from delivery to plate?
How is the walk in and other storage areas organized?
Are the cooks cleaning as they work? In what manner do they do that?
Were there any special customer requests? how did the kitchen handle them?
As Nicko noted, you may not learn advanced techniques, but you can learn much about everything else. Overall, while you are performing simple tasks, your eyes and ears should be wide open, taking in everything around you. Not all three star Michelin places are the same as each other, but they are all operating at a very high level. When your stage is over, you should have a clear picture of how such a place operates and use that knowledge to inform how your place will operate some day and a clear idea of how to conduct yourself from here on out.
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