Cook bacon in heavy large skillet over medium-high heat until brown and crisp; transfer to paper towels to drain.
Wipe out skillet. Brush cut sides of bread with 1 tablespoon oil. Place bread, cut sides down, in skillet. Cook over medium heat until golden, about 3 minutes. Place 1 bread square, golden side up, on each of 2 plates. Top each with half of bacon, then 2 tomato slices.
Whisk 1 tablespoon oil, shallot, and vinegar in medium bowl to blend. Season dressing with salt and pepper. Add arugula and toss to coat.
Heat remaining 1 tablespoon oil in same skillet over medium heat. Crack eggs into skillet. Sprinkle with salt and pepper. Cook until whites are set and yolks are cooked as desired. Top each bread stack with egg, then arugula and cheese.