For the dressing: Stir the mayonnaise, mustard and horseradish together in a small bowl and set aside.
For the burgers: Gently mix together the ground beef, sausage and 1/4 teaspoon each salt and pepper. Divide the meat into 4 pieces and form each into a 4-inch patty. Melt the butter in a 12-inch nonstick skillet over medium-high heat and add the patties. Cook until well browned, about 3 minutes, and then flip. Top with some of the cheese and continue to cook until the cheese is melted and the burger is just cooked through, 2 to 3 minutes. Spread the toasted muffins with some of the mustard dressing and top with a slice of tomato and a burger patty. Crack the eggs into the skillet with the sausage drippings and cook to your preference. Place an egg on each burger and finish with some baby arugula on the top of the English muffin.
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