Preheat the oven to 350 degrees. Arrange 2 racks evenly spaced in the oven.
Place the chicken breasts, skin side up, on one sheet pan. Pat the chicken dry with paper towels, rub the tops completely with olive oil, and sprinkle generously with salt and pepper. Roast for 35 to 40 minutes, until a meat thermometer registers 140 degrees. Cover with aluminum foil and allow to rest for 10 minutes. When cool enough to handle, discard the skin and bones, and cut the chicken in 1-inch dice.
At the same time, place a baking rack on the second sheet pan and lay the bacon out on the rack in one layer. Roast the bacon for 20 to 25 minutes in the same oven with the chicken, until browned and cooked. Set the bacon aside, until cool enough to handle and cut in large dice.
Meanwhile, make the vinaigrette: Whisk together the lemon juice, olive oil, mustard, salt, and pepper in a measuring cup and set aside.
Cut the avocados in half, remove the seeds, and peel. Cut into 1-inch dice, toss with the lemon juice, and set aside.
Place the chicken and tomatoes in a large serving bowl. Sprinkle generously with salt and pepper and enough vinaigrette to moisten. Toss well. Add the avocados, bacon, Stilton, arugula, and additional vinaigrette and toss carefully. Serve at room temperature.