Not if an absolutely discernible mechanism does not exist. For instance, suppose the following day you are placed in the exact same scenario. But on that day you choose food B. And let's suppose this doctor can only view your brain function, and has no idea that there are actually two different types of food. Let's call the the brain activity that occurred when you chose food A, brain function A. If brain function A also occurred on the second day, the doctor would not be able to discern a distinguishable mechanism for you choosing food A, as opposed to food B. It would appear to him you made the same choice.
Let's suppose however, that there is a difference in brain function between the two days, if everything you do on both days is identical up until you make the choice, the doctor would not notice a difference until the choice is made. So on a third day, if you are in an identical scenario as the first two days, there is no way the doctor would be able to determine conclusively which food you will choose until you make the choice. Because brain function A is identical to brain function B up until the choice is made.