Preheat oven to 400°. Place half of the cheese on 8 bread slices; top with ham and remaining bread. Spread outsides of sandwiches with softened butter.
On a griddle, toast sandwiches over medium heat 2-3 minutes on each side or until golden brown. Spread tops with mayonnaise; top with remaining cheese. Transfer to an ungreased baking sheet; bake 4-5 minutes or until cheese is melted.
Meanwhile, for eggs, heat 1 tablespoon butter on griddle over medium-high heat. Break 4 eggs, 1 at a time, onto griddle. Reduce heat to low. Cook to desired doneness, turning after whites are set if desired. Sprinkle with salt and pepper. Place eggs over sandwiches. Repeat with remaining ingredients.
