I was on the cut the burger in half bandwagon for awhile until a friend with two little boys pointed out that she would have problems if one kid got a bigger half.
(Wasn't there a commercial where the solution to that is let one kid cut the burger in half and the other kid make the first choice as to which half they wanted?)
There was also an explanation that some adults liked to pile on the toppings and you can't pile them on half a burger.
That's one solution, and it works. The other is to look at your children, say "fine" - throw the food into the trash.
(For us its never been the park where this has happened - its splitting a can of soda at home. And you remove it from both of them, throw it away, and suddenly they realize that some coke is better than 'more coke than your sister.' )