1/4 cup mayonnaise
1 tablespoon Dijon mustard
1 tablespoon drained prepared horseradish
12 ounces ground beef (80-percent lean)
8 ounces bulk Italian sausage
Kosher salt and freshly ground black pepper
1 tablespoon unsalted butter
1/2 cup shredded Cheddar (about 2 ounces)
4 English muffins, regular, sandwich-size or corn, split and toasted
1 vine-ripe tomato, thickly sliced into 4 pieces
1 cup peppery greens, such as baby arugula, mustard greens or watercress
4 large eggs