Cut the half baguette in half once more, to form two small sub-sized rolls for two sandwiches. Brush two teaspoons olive oil on both side of the two baguettes. Place opened baguettes face down on a large nonstick skillet, cook over medium heat until crispy and lightly golden. Flip over and add cheese slices to each baguette. Allow to cook just until melted. Remove rolls and place on plate.
Add one teaspoon olive oil to the same skillet. Over medium-low heat carefully crack in eggs. Allow to cook for about 3 minutes, until whites are firm but yolks are still a little runny.
Add eggs and arugula to cheese-covered baguettes. Sprinkle with salt and pepper.