1. Preheat oven to 350°. Toss together first 3 ingredients in a medium bowl.
2. Brush melted butter on 1 side of each bread slice. Place bread slices, buttered sides down, on an aluminum foil-lined baking sheet.
3. Bake at 350° for 14 minutes, turning once after 8 minutes.
4. Meanwhile, melt 1 Tbsp. butter in a large nonstick skillet over medium heat. Gently break eggs into hot skillet, and sprinkle with salt and pepper. Cook 2 to 3 minutes or until whites are almost set. Cover, remove from heat, and let stand 1 to 2 minutes or until whites are set and yolks are cooked to desired degree of doneness.
5. Top each bread slice with 2 ham slices and 1 cheese slice. Broil 6 inches from heat 2 minutes or until cheese begins to melt.
6. Top each sandwich with 1/3 to 1/2 cup arugula mixture and 1 fried egg.
Note: For testing purposes only, we used Newman's Own Balsamic Vinaigrette and Boar's Head Black Forest Smoked Ham.