I'm going to assume that you are talking about just cooking and not managing a restaurant or other food business.
The kosher Chinese restaurant in my neighborhood is supervised by the local Va'ad Harabbanim (Rabbinical board.) Nearly all the chefs are non-Jewish Chinese, though I think one might be Jewish (He does speak Hebrew well but with a Chinese dialect. And he wears a "Shmulke Bernstein" yarmulke.)
The place has a mashgiach (supervisor) who is there constantly. The mashgiach or one of his assistants turns on the stove and supervises the deliveries, the food prep (inspects veggies for bugs) and watches the cooking process. One specific item that you may not be allowed to touch is wine. However. most kosher restaurants serve yayin mevushal (boiled wine) which avoids the problem of a non-Jew handling it.
This link, from Star-K, one of the main Orthodox kosher supervising companies in the U.S. explains the rules of bishul akum - cooking by a non-Jew. The mashgiach has multiple duties, as mentioned, above, and he may add his own stringencies in addition to the rules mentioned. It is important to know, that even if the food itself is kosher in its raw or delivered form, bishul akum might make the food non-kosher. The linked article is for information purposes. The mashgiach makes the final decision on what is or is not considered bishul akum, so it's critical to follow his instructions.
In some areas, one may not allowed to eat his 100% kosher lunch anywhere in the restaurant if his lunch is considered chametz or non Kosher for Passover. This may sound strange, and it is rare, but there are caterers, esp. commissaries, that may not serve the public, directly, but prepare packaged food for industrial use, such as airlines or kosher meal plans for hospitals, etc. Passover has an additional restriction to year-round kosher rules in that no leavened products may contact any food. So, the factory may insist that if you eat pastrami on rye for lunch, you must do so outside the factory itself even in January. This way, their products are guaranteed to be "Kosher for Passover" constantly. They do this for two reasons, mainly. 1) Cleaning a factory for Passover is a laborious time-consuming expensive process, and 2) Since these products are packaged and shrink-wrapped and perhaps frozen long term, the consumer / airline, etc. can buy these well in advance of Passover and if there is left-over, they may use it on Passover, as well.
In short, as long as you follow directions of the mashgiach, have fun and cook their food as much as you want!
Please inform me if I understood your scenario to be limited to just cooking using their utensils and not your own.