I would say, strictly by the rules...it's a shady area. The Monk in question has opted to forgo acting on this round to ready an action on the onset of a trigger. Technically speaking, this would then be their action, but is it their turn? This is really the question. Keep in mind, 5e DnD does not have the same concept of turns as previous versions.
If you look at PHB 189, it says that a round is about 6 seconds, and each participate in battle takes a turn in that round. Since the player has opted to ready an action, but not take their turn, then when the trigger occurs, it becomes their turn. Otherwise, if the trigger does not happen during that round, they forfeit their turn.
So the crux comes down to this: Do you think it would be FUN for your players to be able to use their extra attack during a readied attack, or do you think it would make the game less FUN.
Remember, they are there to have fun. If you think it's too OP, and so do your players, then NO they can't. If the players think it's not OP, but you do, then it comes down to DM's choice. Personally, I want the players to have FUN. You can always fudge the numbers on your end if the bad guys are losing too quickly.