Melt the butter in a pan. Sprinkle in the flour and whisk them together for 2-3 minutes.
Mix in the chicken stock and cook in the pan for 4-5 minutes.
Add the mustard sauce and mix until blended. Cover and set aside.
While the gravy is cooking in the pan for 4-5 minutes, pan fry the sausage patties (about 3-5 minutes per side) and start boiling water in a medium sauce pan.
The water should be filled to 1 inch from the top of the pot, with the vinegar and salt in it.
Bring the water to a rolling boil and stir it enough so that the water rotates on its own. Slip the eggs carefully into the water, cover, and cook for five minutes.
Meanwhile, toast the muffins and place the cheese and sausage on the two halves.
At the end of the five minutes for the eggs, remove them from the water immediately with a slotted spoon, setting each one gently atop the patties.
Finish with ever how much gravy you like and you're set to chow down.