Heat the butter in a skillet on medium high temperature until melted, then stir in the flour and cook for 1 minute.
Slowly add the milk while whisking to prevent lumps. Bring to a boil, then add the cheese bits and whisk until melted. Reduce the heat to keep warm. Taste and add salt and pepper as desired, about 1/2 tsp. and 1/4 tsp. respectively.
While making the white sauce, toast the muffins in a toaster. Also, place the asparagus on a dinner sized plate with 2 or 3 tbsp water, and microwave 2 to 2 1/2 minutes until softened.
Bring about 1 1/2 to 2 inches of water to a boil in a tall sided skillet. Once boiling, remove from the heat and gently add the cracked eggs. Cover the skillet for 3 minutes then check to see that the whites are fully opaque and cooked. Remove the eggs with a slotted spoon and place on a plate.
Assemble the dish, first the toasted muffin, asparagus, white sauce and finally the poached egg. Salt and pepper as desired.