Turn the broiler to high and position a rack 4 inches below the heat. Line a baking sheet with foil. Put the English muffins on the prepared pan cut side down and toast under the broiler until firm, just a minute or 2. Turn them over and return to the broiler until they’re browned in spots, another minute or 2.
Put 2 tablespoons butter in a large skillet (preferably nonstick) over medium heat. When the butter has melted, tilt the pan to make sure the butter covers the bottom. Toss the cheese with the mustard and Worcestershire sauce and divide the mixture among the muffins. Return to the broiler just until the cheese is melted and browned in spots, a minute or 2, watching carefully to avoid burning.
When the butter stops foaming, crack 4 eggs into the skillet; it’s fine if they all run into one mass. Cook until the whites just turn opaque, 1 to 2 minutes, then turn the heat down to low and sprinkle the eggs with salt and pepper. Continue cooking until the whites are completely firm, even around the yolk.
Use a spatula to cut between the eggs and carefully transfer them to an English muffin without breaking the yolk. Repeat with the remaining eggs and remaining 2 tablespoons butter. Sprinkle each with salt and pepper, and serve right away.