Preheat oven to 350 degrees F (175 degrees C). Line a rimmed baking sheet with aluminum foil; grease with cooking spray.
Spread chicken thighs open on a flat work surface. Pound to an even thickness with the flat side of a meat mallet. Season with salt and pepper.
Beat egg with a fork in large bowl. Stir in 1/2 cup mozzarella cheese, onion, tomato, and black olives. Mix in enough bread crumbs to bind the filling together.
Put a spoonful of the filling in the center of each chicken thigh. Fold chicken over filling, tucking in any loose ends. Secure chicken thighs with toothpicks; arrange on the prepared baking sheet. Sprinkle remaining 1/4 cup mozzarella cheese over chicken thighs.
Arrange potatoes and carrots around chicken thighs. Drizzle olive oil over vegetables; season with salt and pepper.
Bake in the preheated oven until chicken juices run clear and vegetables are tender, 35 to 40 minutes. An instant-read thermometer inserted into the center of the chicken should read at least 165 degrees F (74 degrees C).