Heat olive oil in a large skillet over medium heat. Add mushrooms; cook and stir until mushrooms are browned and release their moisture, about 5 minutes. Remove from heat, drain, and pat dry.

Beat eggs, sugar, salt, pepper, and red chile powder together in a bowl.

Coat the skillet with some vegetable oil; set over medium heat. Pour in some of the beaten eggs, tilting skillet to spread into a thin layer. Cook until almost set, 1 to 2 minutes. Run a heatproof rubber spatula around the edges to loosen. Cover with some mushrooms; roll up egg and move to the side of the skillet.

Grease skillet with some more vegetable oil. Pour in more eggs to create a second layer. Lift the first roll up to get raw egg underneath. Sprinkle mozzarella cheese on top; roll second layer up over the first.

Repeat layering and rolling process with remaining oil, eggs, mushrooms, and mozzarella cheese. Cook finished tamagoyaki until lightly browned, about 30 seconds per side.