Spread the pesto on the inside of both bagel halves. Set aside.
In a small bowl or mug, scramble the two eggs together with a fork. Season with salt and pepper, to taste. Spray a small skillet with nonstick cooking spray and pour the eggs in the pan. Heat over medium-low heat and cook until just set. Flip and continue to cook until egg is cooked. Place the slice of cheese on top of the egg and turn off the heat. Let the egg sit in the pan for a second so the cheese can melt.
Place some arugula on the bottom half of the bagel and add the cheesy egg on top. Add the avocado slices. Season with salt and pepper, to taste, and add the rest of the arugula. Top with the other bagel half. Serve immediately.
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