Put the eggs in a small saucepan and cover with cold water by 1-inch. Bring to a boil over medium-high heat. Boil for 30 seconds, then take off the heat. Cover and let stand for 8 minutes. Drain and rinse under cold water until cool. Crack all over and then peel.
Put the bacon in a cold skillet and cook, flipping once, over medium heat until crisp, 5 to 6 minutes per side. Transfer the bacon to a paper-towel-lined plate to drain. Increase the heat to medium-high. Sprinkle the chicken lightly with salt and pepper and add to the bacon drippings in the skillet. Cook, flipping once, until golden brown and cooked through, about 4 minutes per side. Transfer to a plate.
Combine the hard-boiled eggs, 1/3 cup water, the oil, vinegar, mustard, 3/4 teaspoon salt, and any accumulated juice from the cooked chicken in a blender and puree until smooth and creamy. Add more water a little at a time to thin if necessary.
Crumble or chop the bacon. Cut the chicken into strips. Toss the lettuce with half of the dressing in a large bowl. Top with the chicken, bacon, avocado, tomatoes, and blue cheese. Drizzle with the remaining dressing.
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