Preheat the oven to 350°F.
Slice the English muffins in half and spread 8 of the halves with the sandwich spread. Spread the other eight halves with the mustard.
Melt the butter over medium high heat on a griddle. Place the English muffins face down (the side with the sandwich spread and mustard) on the griddle and “toast” for 2-3 minutes until golden brown. Remove and keep warm.
Bring a saucepan with 1 quart water to a gentle simmer over medium heat. Add the two tablespoons vinegar. Gently crack the eggs, 2 or 3 at a time, into the water and poach for 3 to 4 minutes. Gently remove with a slotted spoon and place on a plate to drain.
Slice the Braunschweiger into ¼” slices and place on the English muffins. Top with one egg and one slice each of the cheeses.
Place the topped English muffins on a baking sheet and bake until the cheese is melted, about 3 to 4 minutes. Serve immediately.
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