Cook the bacon in a medium skillet over medium heat, stirring, until crisp, 15 to 20 minutes; transfer to paper towels to drain.
Meanwhile, place the eggs in a saucepan and cover with cold water by about 1 inch. Bring to a simmer over medium-high heat, then cover, remove from the heat and let stand, 10 to 12 minutes. Drain, then run under cold water to cool. Peel under cold running water. Chop the hard-cooked eggs and season with salt.
Combine the chicken, lemon zest and juice, bay leaves, thyme, peppercorns and a large pinch of salt in a medium saucepan; add enough cold water to cover the chicken by 1/2 inch. Bring to a bare simmer over medium heat (do not boil) and cook until a thermometer inserted into the thickest part of the chicken registers 160 degrees F, about 7 minutes. Transfer the chicken to a cutting board and let rest 5 minutes.
Meanwhile, make the dressing: Whisk the vinegar, shallot, mustard and 1/2 teaspoon salt in a serving bowl. Whisk in the olive oil in a slow, steady stream until emulsified; season with pepper. Dice the chicken and toss with 1 tablespoon of the dressing in a separate bowl.
Halve, pit and dice the avocados. Season the tomatoes with salt. Add the Bibb and romaine lettuce to the serving bowl on top of the dressing. Arrange the bacon, hard-cooked eggs, chicken, avocados, tomatoes and blue cheese in rows on top of the lettuce. When ready to serve, toss the salad and season with salt and pepper.