I would need to know more about this situation, did the chef just walk out without giving notice? Did the chef give their notice and quit without warning? This plays heavily into the equation here. Usually a resturant will have a "assistant sous chef" or "head line cook" (label changes based on the resturant), that works full time and will preform chef-esque duties and step in if need be. The assistant would most likely step up and act as the "intern sous chef" until management can hire a new sous chef.
I work at a resturant at queens university and we had the sous chef take a 2 week sabatical, and also the night chef walk off the job suddenly. What management did was for the past 2 weeks they didn't use a sous chef during the day (7:30am-4:30pm), instead they relied more heavily on the two food prep workers during the day until the sous chef came back. As for the night chef, because were a big company (the university), we rotated different chefs in each night from different pubs, and for the next week, we are using are sous chef to work both the day and night shift, until we bring on a full time night chef starting on Monday.