I know it's not super original, but I guess hamburgers are a classic American Kid Food for a reason. Kids love them, and they are easy and delicious.
We make these often, but the "buns" are a recent addition. We don't typically eat our hamburgers with buns at home-- they are just served as patties on the plate. I am not sure about the food combining of meat and complex carbohydrates, and it makes for a thick item for kids to hold in their hands. They also really like the plain meat, and I'd rather if they fill their bellies with meat and veggies before anything else.
But on this day, we made the burgers on the Panini Press then got rice bread for buns. We cut it into the right shapes using an upside down cup as a mold/ cookie cutter, then pressed a patty between the slices of bread.
Interestingly, the parts were still eaten separately, but they liked having them served as a unit. It is a nice use of the Panini Press and some meat ~ for variety.