Place potatoes and 2 teaspoons of salt in a large pot and fill with water to cover potatoes. Bring to a boil, reduce to a simmer, and cook for 20 minutes or until tender. Drain and mash with a potato masher. Set aside to cool.Add cheese, green onions, 2 teaspoons salt and ½ teaspoon pepper to potatoes, stirring well to combine. Scoop up about ½ cup of the mixture and form it into a patty that's about 3 inches in diameter and ½ inch thick. Set it on a baking sheet. Repeat with remaining mixture. You should get about 10 patties. Cover and refrigerate patties until firm - for at least ½ hour. (note: you can make them ahead up to this point and keep them in the fridge overnight, if you like)Make the Salad While the potatoes are chilling make the salad. In a small bowl whisk oil, lime juice, garlic, a pinch of salt and a few grinds of black pepper. Place tomatoes, avocado, onion and lettuce (if using) in a bowl. Toss gently with dressing.Cook the Patties: Heat a large cast iron or non-stick skillet over medium heat. Add a tablespoon of oil to the pan and swirl to coat. Cook patties in two batches. Let them to brown on one side for 4-5 minutes, then turn and brown on the other side for 3-4 minutes.Serve patties with tomato-avocado salad. Sprinkle patties and salad with reserved minced green onions.