Cook the chicken: Heat 1 tablespoon olive oil in a large frying pan over medium heat. Pound chicken breasts to even thickness and season with salt and pepper. Cook for about 6 minutes per side until no longer pink inside. Set aside on a cutting board to rest.
Cook the bacon: In the same pan as the chicken, cook the chopped bacon over medium heat until crispy, about 4 minutes, and set aside on a paper towel-lined plate.
Make the dressing: Put vinegar, Dijon mustard, shallots, and olive oil in a jar and shake until well-combined. Season to taste with salt and pepper.
Assemble the salad: Divide the lettuce between 4 plates. Arrange tomatoes, avocados, cheese crumbles, bacon and chicken over the lettuce and drizzle with dressing.