Make dressing: Whisk together all dressing ingredients except oil in a bowl, then add oil in a slow stream, whisking until emulsified.
Make salad: Bring 5 cups water to a simmer in a 2-quart saucepan, then simmer chicken, uncovered, 6 minutes. Remove pan from heat and cover, then let stand until chicken is cooked through, about 15 minutes. Transfer the chicken to a cutting board and cool completely. Cut into 1/2-inch cubes.
Halve pit and peel avocados, then cut into 1/2-inch cubes.
Spread romaine over bottom of a 6- to 8-quart glass bowl and top with an even layer of chicken. Sprinkle bacon over chicken, then continue layering with tomatoes, cheese (to taste), avocados, watercress, eggs, and chives.
Just before serving, pour dressing over salad and toss.
Dressing can be made and salad assembled 1 hour ahead and chilled separately, covered.