Heat oven to 400° F. Rub the chicken with 1 tablespoon of the oil and season with ½ teaspoon salt and ¼ teaspoon pepper. Place on a rimmed baking sheet and roast until cooked through, 20 to 25 minutes. Let cool, then shred.
In a large bowl, toss the romaine with the vinegar and the remaining 2 tablespoons of oil; season with ¼ teaspoon each salt and pepper. Top with the eggs, avocado, bacon, blue cheese, and chicken.