To cook the bacon, heat a large (12-inch) sauté pan over medium heat. Add the bacon and cook for 5 to 7 minutes, turning occasionally, until nicely browned. Transfer to a plate lined with paper towels and set aside.
Meanwhile, generously brush two small microwavable bowls (about 4 inches in diameter) with olive oil and set aside. In a separate bowl, beat together the eggs, milk, ½ teaspoon salt, and ¼ teaspoon pepper. Pour half of the egg mixture into each of the two oiled bowls and microwave on high for 30 seconds. Stir the eggs gently with a fork to combine the cooked parts with the uncooked parts. Continue to microwave and stir at 30-second intervals, until the eggs are puffed and almost cooked through. Sprinkle 1 tablespoon of the Cheddar in each bowl and microwave for 30 seconds to melt the cheese and finish cooking the eggs.
Meanwhile, toast the English muffins until nicely browned. Place the bottom half of each toasted English muffin on a plate. Layer half of the avocado slices on each one and sprinkle with salt and pepper. Break the bacon strips in half and place two halves on each sandwich. Slide the eggs, cheese side up, on top of the bacon, and sprinkle a tablespoon of the remaining Cheddar on each sandwich. Sprinkle with salt and pepper and cover with the tops of the toasted English muffins. Serve hot.
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