For the salsa: Heat oil in a medium saucepan over medium heat. When it shimmers, add tomatillos, onions, jalapeno, and garlic, and cook until tomatillos and onions are softened, about 7 minutes. Transfer mixture to a blender, add cilantro, vinegar, and salt, and puree until sauce is smooth. (Yields about 2 cups salsa.)
For the eggs: Bring at least 3 inches of water to a simmer in a medium pot and stir in vinegar. Break each egg into a separate small cup. Fill a large bowl with warm water and set aside. Gently slide eggs into the simmering water, 1 at a time. Cook until whites are just set, about 2 to 3 minutes. Lift eggs out of the water with a slotted spoon. Place in a bowl of warm water to keep warm while finishing the dish.
For the tortillas: Heat oil in a large heavy bottomed pan over medium-high heat to 350 degrees F. When oil is ready, add tortillas 1 at a time and fry until golden brown and crisp on both sides, about 1 minute. Transfer to a paper-towel-lined plate and season immediately with salt.
To serve: For each serving, place down 2 corn tortillas, place 2 eggs on top, drizzle with salsa, and sprinkle with cheese. If desired, top with a dollop of sour cream and slices of avocado. Serve immediately.