Cut potatoes into cubes, and boil until fully cooked and tender, about 10 minutes. Drain off excess water and transfer potatoes into a mixing bowl. Mash the potatoes lightly using a fork, leaving a some big chunks of potato. Season with Japanese mayonnaise and rice wine vinegar, and add 2 tbsp of the green onions. Mix well and set aside.

To make the white sauce, in a saucepan, melt the unsalted butter over medium heat. Stir in the all purpose flour, making a roux and mix until there is no more visible raw flour. Pour in the milk and stir well. Let cook over medium heat while stirring occasionally until thick, about 10 minutes. Season with garlic powder and salt and pepper to taste.

Lightly grease two heat-proof bowls with oil (we used 10 cm diameter bowls). Scoop about 1/4 of the potato mix onto one bowl and spread it evenly to make a flat bed of potatoes. Pour about 5 tbsp of the white sauce on top of it. Scoop in another 1/4 of the potato mix and place on top of the white sauce. Add another 5 tbsp of the white sauce on top, and finish with 5 tbsp of cheese, distributing them evenly on top of the dish. Repeat the steps for the other bowl. Bake at 375 degrees Fahrenheit for 15 minutes, until cheese are melted.