Add warm water to a large mixing bowl. Sprinkle yeast and sugar over the top. Let the yeast mixture activate for 10 minutes. Add in the flour, salt, and olive oil. Mix together until dough forms. Turn dough out onto clean, floured surface. Knead with your hands until soft, smooth dough forms, at least 6 minutes. Add more flour if dough sticks to the surface. Once dough is smooth and elastic, place into a greased bowl to rise for 1 ½ hours or until it has doubled in size.
While dough is rising, make filling. Place cream cheese in a medium bowl. Microwave for 10 seconds to soften it. Add chicken, wing sauce, ranch dressing, and mozzarella cheese to bowl. Stir together until combined. Set aside for later use.
Preheat oven to 400 degrees. Prepare baking sheet by lining with parchment paper.
Once dough has risen, punch it down and remove from bowl. Place onto clean counter and cut into 6 even wedges. Using a rolling pin, roll 1 wedge into a circle. Add ⅓ cup of filling to the center. Pull one edge of the dough over the center to seal the pocket. Using a fork, seal the edges. You can also roll the edges to make the seal extra-tight. Place pocket onto prepared baking sheet. Repeat with the 5 remaining wedges so that you have 6 filled pockets.
Beat egg with 1 teaspoon of water to make egg wash. Brush egg wash over each pocket. Cut three slits in the center of each pocket to let the steam release while baking. Bake pockets in preheated oven and bake until tops are golden brown, about 35-40 minutes. Remove from oven and let them cool for 5 minutes before serving. Serve with ranch dressing for dipping.