1. Cut bread slices into 8 rounds with a 3 3/4-inch cookie cutter. Top half with a few slices of cheese; sprinkle with oregano, salt, and pepper. Place remaining bread rounds on top; press gently to adhere.

2. In a small bowl, whisk together the milk and 2 eggs. Heat 2 tablespoons oil in a large nonstick skillet over medium heat. Dip sandwiches, one at a time, into egg mixture, letting excess drip back into bowl; transfer to skillet. Cook until golden and cheese has melted, about 2 minutes per side. Transfer to serving plates.

3. Wipe skillet with a paper towel. Heat remaining 2 tablespoons oil. Crack remaining 4 eggs into skillet, one at a time. Fry until whites are just set, about 2 minutes. Carefully place an egg on each sandwich. Season with salt and pepper; garnish with oregano, and serve.