Toast the bagels to your desired doneness. Prepare the avocado, tomatoes, and arugula. Set aside while you cook the eggs.

Add the olive oil to a small frying pan, and heat to medium low heat. Crack an egg into the skillet. Cover with a tight lid and cook, uninterrupted, until the whites are completely set but the yolks are still runny, 2 to 2 1/2 minutes. Slide the eggs out of the skillet. Season with salt and pepper. Repeat with the other eggs.

Assemble each sandwich by spread 1 tablespoon of mayo onto the bottom half of each toasted bagel. Divide the bacon, tomatoes, avocado, arugula and eggs per sandwich. Sprinkle with salt and pepper to taste, then top each bagel sandwich with the top half of the toasted bagel. Serve immediately.