Put mozzarella cheese 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 3o more seconds until uniform and gloopy (it should look like cheese fondue at this point, if it doesn't keep microwaving in 30 second increments). Mix in the egg and baking mix. You will need to dump it onto wax paper and knead it by hand to thoroughly incorporate the ingredients or you can do this in a food processor with the dough blade.
Line an old cookie sheet (make sure it will fit on your grill) with parchment paper. Dump the dough onto the parchment. Press or roll into a large circle on a pizza stone using wet hands or a wet rolling pin.
Put the cookie sheet on the grates. Grill for about 5-10 minutes until the dough has firmed up. The heat of grills varies a lot so it's best to keep peeking at it every few minutes. Once the dough has firmed up and the edges are golden remove the baking sheet from the grill. Wear oven mitts, it is very hot. Flip the dough upside down right on the grates and peel off the parchment. If you want your toppings hot put them on now. Grill for an additional 5 minutes until the dough is fully cooked.