Heat a large non-stick pan over medium with the butter.
Once the butter is melted, add both English muffin pieces face-side down onto one side of the pan. Use your hand or a spatula to move them in a small circle and then press down to ensure the muffins will get crispy. Toast for 1 - 2 minutes until they reach your desired crispy-ness. Alternatively, you can toast your English muffin in a toaster and butter it after.
Meanwhile, whisk the egg(s), salt, and pepper in a small bowl.
Once the English muffin is done, pour the egg mixture into the other side of the buttered pan. Once the eggs just begin to set, use a rubber spatula to gently push them across the pan. Try not to stir, or else they will not be as fluffy. If the eggs cook too fast, reduce the heat to low.
Assemble your breakfast sandwich by placing your soft scrambled eggs on top of the bottom half of the English muffin, followed by the slice of cheese and then the top half. The heat from the eggs will melt the cheese.
Cheese: Use any type of slices of cheese. I love using ex-sharp cheddar cracker cut slices. This means they are a bit smaller but thicker. I use two squares and it yields super cheese, delicious eggs.Adding Bacon or Turkey: Make sure the bacon is pre-cooked and add during step 5 when you are assembling the sandwich.If you are toasting the English muffins in a toaster oven, place the slice of cheese on one side of the muffin in the last minute to make sure it is extra melty.