Yes, that's an egg in this everything bagel. And when you cut into it, yolky yellow (like California sunshine) drips down into the bacon, lettuce, tomato and avocado mash. It's OK if you need a moment.
Preheat the oven to 400 degrees F. Arrange the bacon in a large ovenproof skillet and cook over medium-high heat until crispy, about 4 minutes per side. Transfer to a paper-towel-lined plate. Remove all but 2 teaspoons bacon drippings from the skillet.
Reduce the heat to medium. Cut each bagel in half crosswise. Toast the bottom pieces in the skillet cut-side down until light golden brown, 3 to 4 minutes.
Meanwhile, mash together the avocado, lemon juice, red pepper flakes, 1/4 teaspoon salt and a few grinds of pepper in a small bowl.
Remove the bagel bottoms from the skillet and evenly spread the avocado mash over the toasted sides; set aside. Put the 2 top bagel pieces in the skillet cut-side down. Crack 1 egg into each bagel hole and season with salt and pepper. The whites will run over and under the bagels which is okay-- just make sure the yolks stay in the holes.
Cover the skillet and cook until the whites become crispy and the bottoms of the bagels are toasted, 3 to 4 minutes. Transfer the skillet to the oven and continue cooking until the whites are completely set and the yolk is still runny, about 5 minutes.
Meanwhile, layer each avocado bagel bottom with a slice of tomato, some arugula and 2 slices of bacon. Drizzle with the oil and sprinkle with salt and pepper. Top with the egged bagel tops. Cut each sandwich in half.
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