Place 1 1/2 teaspoons salt and potatoes in a saucepan, and cover with water. Bring to a boil; reduce heat, and simmer 15 minutes or until tender. Drain, and mash with a potato masher until smooth. Cool.
Add cheese, green onions, 1/4 teaspoon salt, and pepper to potato mixture, stirring well. Divide potato mixture into 6 balls (about 1/2 cup per ball). Flatten balls into 1/2-inch-thick patties (about 3-inch diameter). Place on a baking sheet; cover and refrigerate 20 minutes or until firm.
Heat oil in a large nonstick skillet over medium heat. Place potato-and-cheese patties in pan; cook 5 minutes or until bottoms are browned. Turn patties; cook 3 minutes. Top patties with tomato and red onion.