Make the homemade tomato sauce: place the tomatoes, onion, garlic, and ½ cup water in a medium-sized saucepan. Cook over medium heat until the tomatoes burst, about 7-8 minutes.
Transfer to a blender, add ½ teaspoon salt, ½ teaspoon oregano, black pepper to taste, and blend until smooth. In the same saucepan, heat ½ tablespoon olive oil and add the sauce. Bring to a boil, reduce the heat, and simmer for 8-10 minutes. Set aside.
While the sauce is cooking, place the chayote slices on a cutting board and sprinkle with them salt and pepper. Let them rest for 5 mins to release moisture, then pat dry with a paper towel. Make sandwiches using 2 slices of chayote with one slice of cheese in the middle. Sprinkle dry oregano over the cheese.
Beat the egg whites until soft peaks form. Add the egg yolks and whisk again until frothy with soft peaks.
Heat 1 ½ tablespoon of olive oil in a skillet over medium-how heat. While the oil is heating up, dip each sandwich into the egg mixture, then transfer to the skillet.
Pan-fry the sandwiches on one side until golden brown, about 3-4 minutes. Flip, and cook on the other side for an additional 3-4 minutes. Keep the heat medium-low so the egg coating doesn't burn.
Serve over the freshly made tomato sauce with a drizzle of sour cream on top. I added crushed pork rinds for crunch.