How to Make It
Preheat the oven to 300°. Lay the bread slices on a work surface. Arrange the ham and cheese on 4 slices, then close the sandwiches.
In a large skillet, melt 1 tablespoon of the butter in 1 tablespoon of the oil. Brush the sandwiches on 1 side with the melted butter and oil, then add to the skillet, buttered side up. Cook over moderate heat, turning once, until the cheese is melted and the sandwiches are golden, 4 minutes. Transfer to a baking sheet and keep warm in the oven.
Melt the remaining 1 tablespoon of butter in 1 tablespoon of the olive oil in the skillet. Crack in the eggs and cook over moderate heat until the whites are set and the yolks are still runny, about 2 minutes. Top each sandwich with an egg and season with salt and pepper.
In a medium bowl, toss the arugula with the remaining 1 tablespoon of olive oil and the lemon juice and season with salt and pepper. Transfer the sandwiches and salad to plates and serve.