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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