Preheat the oven to 400. Line a cookie sheet with parchment paper.
Put the mozzarella and cream cheese in a microwave-safe bowl. Microwave one minute. Stir. Microwave 30 seconds. Stir. At this point, all the cheese should be melted. Microwave 30 more seconds until uniform and gloopy (it should look like cheese fondue at this point). Add the rest of the dough ingredients and the cheese to a food processor. Mix using the dough blade until a uniform color. If you do not have a food processor you can mix in a medium bowl with a wooden spoon but you may need to dump it onto wax paper and knead it by hand to thoroughly incorporate the ingredients. Press out into a large rectangle and cut into the same number of pieces as pieces of chicken.
To assemble put the pieces of chicken in one bowl, add the beaten egg and stir so they all get coated in egg. Combine the cheeses and seasonings in another bowl. Take one piece of chicken and roll it around in the cheese onion mixture. Take one piece of dough and wrap it around the chicken. Put on the cookie sheet. (Don't worry if they look messy, you can't tell once they are baked. It helps to keep your fingers moistened so the dough doesn't stick to them). Repeat. Once they are all done brush with a little water and sprinkle with sesame seeds.
Bake for 25-30 minutes until the dough is golden and the chicken is fully cooked.
To make the dips: stir together all the ingredients in a medium bowl.