The quotation marks around 'when' indicate that the sentence literally means "The cheese keeps coming until you say the word 'when'". I'm sure that waitstaff would stop dispensing cheese in response to many other possible utterances, but I think the phraseology is intended to avoid any implication that the restaurant would keep dispensing cheese until customers demanded that they stop, or that the restaurant would dispense cheese until the customers were willing to concede that they couldn't reasonably demand more.
The "say 'when'" idiom treats the quantity of cheese as something akin to the volume adjustment on the sound system for a reception or other such event. The sound engineer may ask the client to stand in the middle of the venue, and then increase the volume until the client indicates it's satisfactory, but the sound engineer would hardly view the continued rotation of the volume knob as any kind of burden. Rather, the task would be to find whatever volume setting would best please the customer.
Likewise, the implication with Olive Garden would be to suggest that their goal was to establish whatever balance between cheese and other ingredients would best suit the customer's palate, even for customers where the balance would tilt heavily toward the "cheese" side, and that Olive Garden had no desire to give any customer more or less cheese than the customer would want.
I think the notion of "say 'when'" rather than "say 'stop'" is to avoid any perceived need for the customer to precede an imperative with "please". A customer may respond to a question like "Would you like cream" with "Yes, please" or "No, thank you", but not "Please yes". The response to "Say 'when'" is a response to an implied question, rather than an imperative, thus not requiring a preceding "please". A polite customer may respond to "Say when" with "Thank you" as soon as a suitable amount has been dispensed, or may say "When", and follow that with "Thank you" after dispensing ceases.
Incidentally, the "say when" construct also implies that the waiter intends to be 'directly controllable' in other ways without need for social niceties. Saying 'slower, please' or 'more toward the shrimp, please' would be preferable to 'Could you please slow down a little bit?' or 'Could you please shift more toward the shrimp?'