Melt butter in a 10" skillet over MEDIUM-LOW heat. Crack eggs into skillet and place cover on skillet. Cook for 2 to 3 minutes, or until eggs are set to your liking. Season eggs with salt and pepper, to taste. Remove from heat and set aside.
Meanwhile, preheat your oven to BROIL.Separate English muffins and place the halves on a baking sheet. Place corned beef on bottom halves and then layer with cooked eggs and shredded Irish white cheddar. Place the baking sheet in the oven, watching closely, and allow to BROIL for just a minute or two until muffins are lightly toasted and cheese has melted. Remove from oven.
Add a dollop of Thousand Island dressing to each sandwich and serve.