Combine eggs and spinach in a blender. Blend on high speed for 1 minute. Scrape down the sides of the blender. Pulse until spinach is fully blended into the eggs, about 1 minute more.
Pour hash into a microwave-safe bowl. Heat in the microwave until thoroughly warmed, about 1 minute. Season with black pepper.
Spray a nonstick skillet with cooking spray; place over medium heat. Pour in egg-spinach mixture. Cover with a lid. Cook until top is barely set, about 5 minutes. Place corned beef hash over 1 side of the omelet. Slide onto a plate, using the skillet to fold the other side on top. Sprinkle mozzarella cheese over omelet.