Binders for skillet rice patties
You want to use egg whites for binding. 

Science notes: 

About 1/20,000* eggs has salmonella. For infants &amp; toddlers I would suggest cooking thoroughly or pasteurizing the eggs with sous vide if they won't be fully cooked. 

Egg whites comprise water (88%) and protein (11%, mostly albumin). Egg yolks are water (48%), protein (17%), and fat (33%). (There are some other things that make up the full 100% of each.) The yolk makes up about 33% of the egg.  

The chalazae (the "gross" stringy parts of the egg white that holds the yolk away from the shell) is made from the same protein as the rest of the white. It may look weird but it tastes exactly the same. 

*this is the number estimated by the CDC.