First off... I am DARING to make assumptions based on the information you have here- so if I am off base on your daughter's diet... it is not "on purpose"
She is really too L. to go on a "diet" without specific instructions from a pediatrician... so try not to limit HOW MUCH she can eat- but do start enforcing what she is eating. "carbs" are not good for anyone if they are eaten too much, especially sugar, sugar substitutes, and starches... Try to make sure she has plenty of healthy protein (don't worry about "fat" at this age... kids need "fat" for healthy brain function and growth).
If she is hungry - let her eat- but don't feed her plain carbs automatically... she'll burn through them too fast and be hungry again- and after a while this could lead to high blood sugar (and high blood sugar leads to diabetes- eek! But don't panic about it- this is just a good reason to take action to re-organize her diet now rather than later)
I am not saying to cut out carbs completely- but when she wants a "snack" it may be better to offer some scrambled eggs... a slice of cheese, or some plain yogurt with fruit (which have healthier carbs).
Stop bringing donuts home, and try to offer more balanced alternatives for breakfast- with a combination of carbs and proteins- an egg sandwich (on whole grain toast or an english muffin) a with 1 slice of bacon or sausage may not sound "healthy"- but for a kid it has the carbs to get their metabolism going, and the protein to give their breakfast "staying power" through lunch. A slice of cheese would hurt if that isn't enough for her.
Her body needs to reprogram- she needs foods that will help her grow "up" rather than "out"... which are muscle and brain building foods like healthy fats and proteins. She also needs lots of vitamins and nutrients... so consider amping up her supply of "powerfoods" (think blueberries, prunes, carrots, spinach... etc) - bright, naturally "vibrant" foods often have the most nutrients--- naturally "dull" colored foods often have less.
Of course... if you can, consider making a food log of all she is eating for a week... and then take it in to the pediatrician or a child-nutritionist. They may be able to look at that log and say exactly what foods your daughter has too much of, and what she is missing out on. Your daughter is hungry because her body NEEDS something... but only by seeing what she is missing can you discover what that "something" is...
And then relax... plenty of kids are chunky at stages in their lives... especially right before they "shoot up" in height. Of course some kids are skinny bean-poles all their lives and then plump out the second they hit adulthood! As long as she is still growing, has plenty of energy, and is developing normal you probably have L. to worry about)
Good Luck!
-M.
oh- and the QP question... if she wants the burger... get her the burger- but for a drink insist that she has milk, or even an iced tea... and skip the fries or stick to a small or "value" order.... the sides are what "git" ya- not the burger! BUT that is if she goes there often... if a trip to a fast food place is once in a blue moon (like once a month) let her have what she wants, so long as she can eat it. This is because it is a treat- and you have changed her eating pattern at home... right? Also- since you said that she stays with mom 50% of the time... try to arrange the same diet changes there too... you cannot MAKE them do anything "over there"- but chances are they can see eye to eye with you so long as it is in the best interests of your daughter!