Wash the potatoes and dry with kitchen towel. Peel off the skin and cut potatoes into small cubes of approximatively the same size. In a large pot, bring water to a boil and add 1 teaspoon of salt. Gently place the potato cubes into boiling water and make sure all the potatoes are submerged in the water. Boil at medium high heat for about 9-10 minutes or until soft and tender. Remove potatoes from water and drain off excess water.
In a large bowl, place the cooked potatoes and mash using fork until smooth. Add the sliced carrots, sliced cucumbers, chopped green onions, and sticks of red onions into the bowl. Add the salt, vinegar, and Japanese mayonnaise and combine everything together until well mixed.
Once the mashed potatoes are cooled, form about 1 1/2 to 2 heaping tablespoons of mashed potato into a flattened oval ball. Repeat with the rest of the mashed potatoes (the recipe should yield about 5-6 croquettes). Place croquette balls on a plate and let chill in the refrigerator for about 30 minutes.
Once the croquettes are cooled and firm, coat each ball with all-purpose flour, then egg, and finally Panko breadcrumbs. Repeat these steps with each croquette ball.
Heat about 1/2 inch of oil in a pan on high heat. Once the oil is hot enough, bring temperature to medium high and fry the croquettes for about 2-3 minutes on each side or until golden brown. Remove from heat and drain off excess oil.