Toast the English muffins in the toaster, in batches, or in the toaster oven. During this process, start cooking the eggs.
In a bowl, whisk the eggs.
Heat a large non-stick pan over medium-high heat. Spray or grease with oil, and once hot, pour the whisked eggs into the pan. Cook the eggs, either scramble them or let the egg mixture set lightly before breaking it it apart, and with a spatula, flip them over into a large omelet. It often breaks apart but that's ok!
Place the toasted English muffins on a baking sheet middle-side up. Top 6 halves with a slice of cheese and the other 6 with a slice of ham (fold it if needed).
Place it in the oven, and BROIL it for 1 minute, until the cheese has melted and the ham has warmed up. Remove the baking sheet from the oven.
Distribute the cooked eggs on top of the warm ham slices and close the breakfast sandwich with the melted cheese side.