You can complete a menu anytime that it is your turn to do something - whether it's placing a chef cap pawn or purchasing an ingredient. Basically, while the spotlight is on you to perform an action, you can announce that you're also going to complete a menu, then do so before the next player can take their turn to do something.
Hope this helps!
This raises another question. does the action "Completing a menu" replaces whatever action you could've done? For example, after I purchase an ingredient such that now I can complete a menu, do I have to wait for my next turn to complete the menu or right after purchasing an ingredient I can complete the menu right away?
Similarly, if during my turn I already have all the ingredients for a menu at hand and I announce I'm completing a menu, then after completing it, can I still buy an ingredient?