In a large skillet over low heat, combine 2 tablespoons olive oil and butter; cook until melted.
Meanwhile, whisk together milk, eggs, and Parmesan and season with salt and pepper. Dip the bread slices in the egg mixture. Don’t let the slices soak too long.
Increase skillet heat to medium-high and add the bread slices and cook until browned on both sides. Remove from pan and keep warm.
Meanwhile, toss the arugula, remaining olive oil, and ham together and season with salt and pepper. When the bread is toasted, place on a plate and top with the salad. Garnish with Parmesan cheese, and, for extra pizzazz, top with a sunny-side up egg.