Grilled Ham and Cheese Sandwiches with Fried Eggs

Directions

1. 1
   Preheat the oven to 300.
2. 2
   Lay the bread slices on a work surface.
3. 3
   Arrange the ham and cheese on 4 slices, then close the sandwiches.
4. 4
   In a large skillet, melt 1 tablespoon of the butter in 1 tablespoon of the oil.
5. 5
   Brush the sandwiches on 1 side with the melted butter and oil, then add to the skillet, buttered side up.
6. 6
   Cook over moderate heat, turning once, until the cheese is melted and the sandwiches are golden, 4 minutes.
7. 7
   Transfer to a baking sheet and keep warm in the oven.
8. 8
   Melt the remaining 1 tablespoon of butter in 1 tablespoon of the olive oil in the skillet.
9. 9
   Crack in the eggs and cook over moderate heat until the whites are set and the yolks are still runny, about 2 minutes.
10. 10
    Top each sandwich with an egg and season with salt and pepper.
11. 11
    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.
12. 12
    Transfer the sandwiches and salad to plates and serve.