Split the English muffins in half and toast them to your desired crispness.
To make the Mornay sauce, melt the butter in a small saucepan over low heat.
Whisk in the flour to form a paste (a roux) and cook for 3 minutes, stirring continually to cook out the raw flour taste.
Gradually whisk in the milk until the mixture is smooth and begins to thicken.
Remove from the heat and whisk in the shredded Parmesan cheese until it has completely melted.
Stir in the prepared mustard, ground nutmeg, and a pinch of salt and pepper.
To assemble, place one poached egg on one half of a toasted English muffin. Spoon a quarter of the warm Mornay sauce over the other half, then close the muffin to create a sandwich.
Repeat with the remaining muffins and serve immediately.
Add a pinch of cayenne pepper or a dash of your favorite hot sauce to the Mornay for a subtle, warming kick of heat.