Position an oven rack 5 inches from the heat and preheat the broiler to high.
Using a damp kitchen towel, clean mushrooms. Remove stems, and use a spoon to scrape out and discard dark gills.
Place mushrooms, flat side up, on a rimmed baking sheet and brush them with Worcestershire.
Place baking sheet in oven and broil for 2 minutes.
Flip mushrooms and broil for 2 minutes more, or until they are tender and browning. Remove from oven and sprinkle with salt and pepper.
Place 2 mushroom caps on each of 4 toasted muffin bottoms and cover with aluminum foil to keep warm.
In a medium nonstick skillet, heat1⁄2 teaspoon of the oil over high heat.
Crack 2 eggs into skillet. Cook for 1 minute or until whites are set and bottom is beginning to brown, and then flip and cook for 30 seconds more, or longer if you don’t like your yolks runny.
Remove them from skillet and repeat with remaining 1⁄2 teaspoon oil and remaining eggs.
Transfer cooked eggs to mushroom burgers and serve topped with tomato, lettuce, ketchup, and English muffin tops.