1. Preheat panini grill, or place grill pan over medium-high heat. Combine goat cheese and mozzarella in a medium bowl; set aside.
2. Heat oil in a large nonstick skillet over medium heat. Gently break eggs into hot skillet; sprinkle with salt and pepper. Cook 2 minutes on each side or until desired degree of doneness.
3. Spread reserved cheese mixture on 1 side of 2 bread slices, and top with bell pepper and eggs. (You will have 2 eggs per sandwich.) Top with arugula and remaining bread slices.
4. Place sandwiches on panini grill or grill pan; cook 3 to 4 minutes or until golden. Let stand 3 minutes. Cut each sandwich in half with a serrated knife.